Regents Physics Study Guide — Mr. Porter's Physics

1. Measurement, units & math tools

Vectors and scalars

A vector is a physical quantity that has both a magnitude (size) and a direction. Examples: displacement, velocity, acceleration, force, momentum.
A scalar is a physical quantity that has only a magnitude. Examples: distance, speed, mass, time, energy.
A vector can be resolved into perpendicular components. The resultant of two or more vectors is found by vector addition.

Components of any vector A at angle θ from the horizontal

\[ A_x = A\cos\theta \qquad A_y = A\sin\theta \]

Vector components the horizontal component equals A times the cosine of the angle; the vertical component equals A times the sine of the angle.

Diagram — resolving a vector into components

A vector A (the rust arrow) drawn from the origin up and to the right at angle θ above the horizontal. It is the hypotenuse of a right triangle. The horizontal leg along the x-axis is the horizontal component A_x = A cos θ; the vertical leg is the vertical component A_y = A sin θ. The two components meet at a right angle, and together they add (tip-to-tail) to give the original vector.

To get the magnitude of a resultant from two perpendicular components, use the Pythagorean theorem (see math tools below). Perpendicular velocities and perpendicular forces both add this way.

The metric system & SI units

Physics uses SI units. The fundamental units you must know are the meter (m) for length, the kilogram (kg) for mass, and the second (s) for time. Other units (newton, joule, watt, volt, and so on) are built from these.

Prefixes rename powers of ten — for example, 1 kilometer is 10³ meters and 1 millimeter is 10⁻³ meters. The full prefix list is in the reference data tables.

Geometry & trigonometry you need

These appear on the reference tables and are used throughout the course.

\[ A_{\text{rectangle}} = bh \qquad A_{\text{triangle}} = \tfrac{1}{2}bh \]

Areas area of a rectangle equals base times height; area of a triangle equals one-half base times height.

\[ A_{\text{circle}} = \pi r^2 \qquad C = 2\pi r \]

Circle area of a circle equals pi times the radius squared; circumference equals two pi times the radius.

\[ c^2 = a^2 + b^2 \]

Pythagorean theorem for a right triangle, the hypotenuse squared equals the sum of the squares of the two legs.

\[ \sin\theta = \frac{a}{c} \qquad \cos\theta = \frac{b}{c} \qquad \tan\theta = \frac{a}{b} \]

Right-triangle trig in a right triangle, the sine of an angle equals the opposite side over the hypotenuse; the cosine equals the adjacent side over the hypotenuse; the tangent equals the opposite side over the adjacent side.

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2. Kinematics (motion)

Definitions

Displacement (d) tells how far an object ends up from its starting position, regardless of the total distance it travelled. Displacement is a vector; distance is a scalar.
Average velocity (v-bar) is displacement divided by the time interval over which it happened.
Instantaneous velocity (v) is how fast an object is moving at one specific moment.
Average speed is total distance divided by total time.

\[ \bar v = \frac{d}{t} \]

Average velocity average velocity equals displacement divided by time.

Position–time graphs

How far the object is from the detector: read the value on the vertical axis.
How fast the object is moving: look at the slope (steepness) of the line. For a curved graph, take the slope of a tangent line to get instantaneous speed.
Which way the object moves: look at the direction of the slope. A line sloping up like a forward slash means moving away from the detector; a line sloping down like a backslash means moving toward the detector.
Curvature shows acceleration. Concave up (a U shape) means positive acceleration; concave down (a frown shape) means negative acceleration.

Diagram — the four basic position-time shapes

Four position-time curves for uniform acceleration (position d on the vertical axis, time t on the horizontal axis). Described in words:

Speeding up in the positive direction: a curve that starts nearly flat and bends upward more and more steeply (concave up, positive slope getting steeper).
Slowing down in the positive direction: a curve that rises steeply then flattens out (concave down, positive slope getting gentler).
Speeding up in the negative direction: a curve that starts nearly flat and bends downward more and more steeply (concave down, negative slope getting steeper).
Slowing down in the negative direction: a curve that falls steeply then flattens out (concave up, negative slope getting gentler).

These four shapes can be combined to describe more complicated motion.

Velocity–time graphs

How fast (speed): read the value on the vertical axis.
Which way: above the horizontal axis means moving away from the detector; below the axis means moving toward the detector.
How far (displacement): find the area between the line and the horizontal axis.
Acceleration is the slope of a velocity-time graph.
You cannot find the object’s location from a velocity-time graph.

Diagram — reading a velocity-time graph

Velocity v is on the vertical axis and time t on the horizontal axis. The straight line rising from left to right shows constant acceleration, and the line’s slope is the acceleration. The shaded region between the line and the time axis is the displacement. A line above the axis means moving away from the detector; below the axis means moving toward it.

Acceleration

Acceleration tells how much an object’s velocity changes each second.
When an object speeds up, its acceleration points in the direction of motion. When it slows down, its acceleration points opposite the motion.
In free fall near Earth’s surface, acceleration is g, about 9.81 m/s² (often rounded to 9.8, and to 10 for quick estimates).
Units of acceleration are meters per second squared (m/s², also written m/s/s).

\[ a = \frac{\Delta v}{t} \]

Acceleration acceleration equals the change in velocity divided by time.

Special equations for displacement

At constant speed, displacement is speed times time. This includes the horizontal part of projectile motion.

\[ d = vt \]

Constant speed displacement equals velocity times time.

When an object starts from rest and speeds up, or slows to a stop, use one-half a t squared, or the fuller form below.

\[ d = \tfrac{1}{2}at^2 \qquad d = v_i t + \tfrac{1}{2}at^2 \]

Accelerated displacement displacement equals one-half acceleration times time squared; or displacement equals initial velocity times time plus one-half acceleration times time squared.

Algebraic kinematics — the method

Define a positive direction (usually “away from the detector”) and label it.
State in words which part of the motion you are analyzing, for example “from launch to the peak of the flight.”
Fill out a chart of the five kinematics variables — with signs and units: displacement d, initial velocity v-initial, acceleration a, final velocity v-final, and time t.
If you know three of the five variables, the problem is solvable. Pick the equation that uses what you have.

The three kinematics equations

\[ v_f = v_i + at \]

No displacement final velocity equals initial velocity plus acceleration times time.

\[ d = v_i t + \tfrac{1}{2}at^2 \]

No final velocity displacement equals initial velocity times time plus one-half acceleration times time squared.

\[ v_f^{\,2} = v_i^{\,2} + 2ad \]

No time final velocity squared equals initial velocity squared plus two times acceleration times displacement.

A fourth equation is sometimes handy — it is the area of a velocity-time graph written as the area of a trapezoid:

\[ d = \tfrac{1}{2}t\,(v_i + v_f) \]

No acceleration displacement equals one-half time times the quantity initial velocity plus final velocity.

Projectile motion

For an object in free fall, the vertical acceleration is always g (about 9.81 m/s² downward near Earth).
The horizontal acceleration is always zero, so the only horizontal equation you use is displacement equals velocity times time (d equals v t).
If an object is dropped, its initial velocity is zero.
If launched upward, the speed it returns with at its original height equals the launch speed, but in the opposite direction.
An object thrown upward slows as it rises; its vertical velocity is zero at the very top, so you may treat final velocity as zero at the peak.
Velocities in perpendicular directions add with the Pythagorean theorem, just like perpendicular forces. The magnitude of an object’s velocity is its speed.
Method: make two kinematics charts — one vertical, one horizontal. Time is the quantity they share.

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3. Forces & Newton’s laws (dynamics)

Equilibrium

An object is in equilibrium if it moves in a straight line at constant speed — this includes staying at rest.
When an object is in equilibrium, the forces on it are balanced (the net force is zero).

Newton’s first law (inertia)

An object stays at rest, or keeps moving at constant velocity, unless an unbalanced force acts on it. An object’s inertia — its resistance to a change in motion — is directly proportional to its mass.

Newton’s second law

Except for gravity, objects must be in contact to exert a force on each other.
The net force and the acceleration point in the same direction — the direction in which the forces are unbalanced.

\[ \sum F = ma \qquad\Longleftrightarrow\qquad a = \frac{F_{\text{net}}}{m} \]

Newton’s 2nd law the sum of the forces (the net force) equals mass times acceleration; equivalently, acceleration equals net force divided by mass.

Solving force problems — the two-step process

Draw a free-body diagram. Break any angled forces into components first if needed.
Write one Newton’s-second-law equation for each direction. Start each equation in the direction of acceleration so you never get a “negative force”: up forces minus down forces equals mass times the vertical acceleration; left forces minus right forces equals mass times the horizontal acceleration. If the acceleration is downward, write down forces minus up forces instead.

A free-body diagram includes

A labelled arrow for each force, starting on the object and pointing the way the force acts.
A list of the forces, naming the object applying each force and the object experiencing it.

Diagram — example free-body diagram

A box resting on a surface and being pushed to the right. Four labelled force arrows start on the object and point the way each force acts: F_N the normal force points straight up; F_g the weight points straight down; F_app the applied push points right; and F_f the force of friction points left, opposing the motion. If the box moves at constant velocity the up and down arrows balance and the left and right arrows balance; if any pair is unbalanced, the box accelerates in that direction.

Mass and weight

Mass is how much material is in an object, measured in kilograms (kg). Mass does not change with location.
Weight is the gravitational force a planet exerts on an object, measured in newtons (N).
On Earth’s surface the gravitational field is about 9.81 N/kg (often rounded to 10 N/kg), so 1 kg of mass weighs about 9.81 N.

\[ F_g = mg \qquad\Longleftrightarrow\qquad g = \frac{F_g}{m} \]

Weight weight equals mass times the gravitational field strength; equivalently, the gravitational field strength equals weight divided by mass.

Normal force

A normal force is the force of a surface pushing on an object that touches it. It always acts perpendicular to the surface.
A platform (bathroom) scale reads the normal force.

Components of a diagonal force

When a force’s angle is measured from the horizontal:

\[ F_x = F\cos\theta \qquad F_y = F\sin\theta \]

Force components the horizontal component of the force equals the force times the cosine of the angle; the vertical component equals the force times the sine of the angle.

To find the magnitude (the “amount”) of a resultant force from perpendicular components, use the Pythagorean theorem.

Inclined planes (ramps)

On a ramp, break the object’s weight into components parallel and perpendicular to the incline — do not use horizontal/vertical axes.

\[ F_{\parallel} = mg\sin\theta \qquad F_{\perp} = mg\cos\theta \]

Weight on a ramp the component of weight parallel to the incline equals mass times g times the sine of the ramp angle; the component perpendicular to the incline equals mass times g times the cosine of the ramp angle.

Diagram — forces on an inclined plane

A block rests on a ramp tilted at angle θ from the ground. Three real forces act on it (solid arrows): the weight F_g = mg points straight down; the normal force F_N points away from and perpendicular to the ramp surface; and friction F_f points up along the surface, opposing the slide. The weight is then split into two dotted components along the ramp’s own axes: the parallel component mg sin θ points down the slope (this is what pulls the block down the ramp and is balanced by friction), and the perpendicular component mg cos θ presses into the surface (balanced by the normal force). The dotted rectangle shows that these two components add back to the full weight.

Friction

Friction is the force of a surface on an object, acting along the surface, in the direction opposite the object’s motion relative to that surface.
The coefficient of friction is not a force — it is a number describing how “sticky” two surfaces are.

\[ F_f = \mu F_N \]

Friction the force of friction equals the coefficient of friction times the normal force.

Use the kinetic coefficient when the object is moving and the static coefficient when it is not. Both obey the same equation.
Static friction can take any value up to a maximum. For two given surfaces, the maximum static coefficient is greater than the kinetic coefficient.
Values for common surfaces are in the coefficients-of-friction table.

Newton’s third law

The force of object A on object B is equal in size and opposite in direction to the force of object B on object A.
A third-law force pair is this pair of forces. The two forces in a pair never act on the same object.

Two-body problems

Draw one free-body diagram per object.
Write net force equals mass times acceleration for each object separately.
The two objects share the same acceleration.
One rope means one tension.

Hooke’s law (springs)

The stretch or compression of a spring depends on the spring’s stiffness (its spring constant) and the force applied.

\[ F_s = kx \]

Spring force the force on a spring equals the spring constant times the distance the spring is stretched or compressed from its equilibrium position.

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4. Circular motion & gravitation

Uniform circular motion

An object moving at constant speed in a circle is still accelerating, because its direction keeps changing.
This centripetal acceleration points toward the center of the circle.
Centripetal force is the net force that produces circular motion. It is the sum of all the radial forces (forces along the radius) and points toward the center, perpendicular to the velocity.

Diagram — vectors in uniform circular motion

An object travels clockwise around a circle. The velocity (blue arrow) always points along the tangent to the circle — here, at the top of the loop, it points to the right. The centripetal acceleration and the centripetal force (rust arrow) both point toward the centre, along the radius, at a right angle to the velocity. The speed stays constant; it is the constantly changing direction that requires this inward force.

\[ a_c = \frac{v^2}{r} \]

Centripetal acceleration centripetal acceleration equals speed squared divided by the radius of the circle.

\[ F_c = m a_c \]

Centripetal force centripetal force equals mass times centripetal acceleration.

Universal gravitation

Every object with mass attracts every other object with mass. Gravitational forces are always attractive.
The force follows an inverse-square law: double the distance and the force drops to one-quarter.

\[ F_g = \frac{G m_1 m_2}{r^2} \]

Newton’s law of gravitation the gravitational force equals the universal gravitational constant times the product of the two masses, divided by the distance between their centers squared.

F_g: gravitational force, in newtons
G: universal gravitational constant, 6.67 × 10⁻¹¹ N·m²/kg²
m₁, m₂: the two masses, in kilograms
r: distance between the centers of the two objects, in meters

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5. Work, energy & power

All forms of energy are measured in joules (J).

Forms of energy

\[ KE = \tfrac{1}{2}mv^2 \]

Kinetic energy kinetic energy equals one-half mass times speed squared.

\[ \Delta PE = mg\,\Delta h \]

Gravitational potential energy the change in gravitational potential energy equals mass times g times the change in height. Take height equal to zero at the lowest point of the motion.

\[ PE_s = \tfrac{1}{2}kx^2 \]

Spring potential energy the potential energy stored in a spring equals one-half the spring constant times the stretch or compression squared.

Mechanical energy is the sum of a system’s kinetic and potential energy.
Internal energy (Q) is heat energy that raises the temperature of the system; work done against friction increases it.

\[ E_T = PE + KE + Q \]

Total energy total energy equals potential energy plus kinetic energy plus internal energy.

Work

Positive work is done by a force parallel to the displacement.
Negative work is done by a force opposite (antiparallel) to the displacement.
No work is done by a force perpendicular to the displacement.
Work is a scalar — it can be positive or negative but has no direction.
The area under a force-versus-displacement graph is the work done.

\[ W = Fd = \Delta E_T \]

Work work equals force times displacement, and also equals the change in the total energy of the system.

The work–energy theorem — the method

Define the object or system you are describing.
Define the two positions (start and end) you are comparing.
Draw an annotated energy bar chart.
Write one equation with one term per bar: initial energy plus or minus work equals final energy.

Power

Power is the work done (or energy used) each second.
Units are joules per second, called watts (W).

\[ P = \frac{W}{t} = \frac{Fd}{t} = F\bar v \]

Power power equals work divided by time, which also equals force times displacement divided by time, which also equals force times average velocity.

Energy in collisions

In an elastic collision, the system’s mechanical (kinetic) energy is conserved.
Collisions where objects stick together cannot be elastic (they are inelastic).
Collisions where objects bounce apart may or may not be elastic.

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6. Momentum & impulse

Momentum

Momentum is mass times velocity. Its direction is the same as the direction of motion.
Units are newton-seconds (N·s), equivalent to kilogram-meters per second.

\[ p = mv \]

Momentum momentum equals mass times velocity.

Impulse

Impulse can be found two ways: as the change in an object’s momentum, or as the net force multiplied by the time it acts. Impulse is the area under a force-versus-time graph, and has the same units as momentum (N·s).

\[ J = F_{\text{net}}\,t = \Delta p = m\,\Delta v \]

Impulse–momentum impulse equals net force times time, which equals the change in momentum, which equals mass times the change in velocity.

Conservation of momentum

When no external force acts on a system, its total momentum cannot change.
The total momentum of two objects before a collision equals their total momentum after.
Momentum is a vector: momenta in the same direction add; momenta in opposite directions subtract.
A system’s center of mass obeys Newton’s second law — its velocity changes only when an external net force acts.

\[ p_{\text{before}} = p_{\text{after}} \]

Conservation of momentum total momentum before equals total momentum after.

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7. What is conserved?

Mechanical energy is conserved when no net work is done by external forces (and there is no internal-energy conversion, such as friction heating).
Momentum in a direction is conserved when no net external force acts in that direction.
Mass–energy is conserved — mass can convert to energy and back, related by E equals m c squared.
Charge is conserved — it can be moved or neutralized but not created or destroyed.

\[ E = mc^2 \]

Mass–energy energy equals mass times the speed of light squared.

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8. Waves & sound

Wave anatomy

Crests are the high points; troughs are the low points.
Amplitude (A) is the distance from the midpoint to a crest or trough. It determines the energy the wave carries.
Wavelength (λ) is the distance between identical parts of the wave (for example, crest to crest).
Frequency (f) is the number of waves passing a point each second, measured in hertz (Hz).
Period (T) is the time for one wavelength to pass a point.

Diagram — a transverse wave
A transverse wave drawn as a smooth curve travelling to the right along a dotted horizontal **equilibrium position** line. The high points are **crests** and the low points are **troughs**. The **amplitude (A)** is the vertical distance from the equilibrium line up to a crest. One **wavelength (λ)** is the horizontal distance from one crest to the next, or one trough to the next. An arrow labelled “Direction of Wave Travel” points to the right; the medium itself moves up and down, at right angles to the direction the wave travels.

Key facts

When a wave moves from one material to another, its frequency stays the same.
The speed of a wave depends on the material it travels through.
It is the disturbance that travels, not the material itself.

Relating frequency, period, wavelength, and speed

\[ f = \frac{1}{T} \qquad T = \frac{1}{f} \]

Frequency & period frequency equals one divided by the period; the period equals one divided by the frequency.

\[ v = f\lambda \qquad v = \frac{\lambda}{T} \]

Wave speed wave speed equals frequency times wavelength; it also equals wavelength divided by the period.

Transverse and longitudinal waves

In a transverse wave the material vibrates at right angles to the direction the wave travels. Waves on a string and all electromagnetic waves (including light) are transverse.
In a longitudinal wave the material vibrates parallel to the wave’s direction. Sound waves are longitudinal.
All electromagnetic waves travel at 3.0 × 10⁸ m/s in air or vacuum. Light can travel through a vacuum; sound cannot.
The higher the tension in a string, the faster a wave moves along it.

Diagram — a longitudinal wave
A longitudinal wave shown on a horizontal spring (slinky) stretched between two **fixed ends**. The coils bunch together in **compressions** and spread apart in **rarefactions**. One **wavelength (λ)** is the distance from one compression to the next. Arrows for “Direction of Wave Travel” and “Energy Transfer” point to the right, while individual particles oscillate *back and forth parallel* to that direction, about their equilibrium position. **Sound** is a longitudinal wave.

The electromagnetic spectrum

Visible light is one small slice of the electromagnetic spectrum. All electromagnetic waves travel at 3.00 × 10⁸ m/s in a vacuum; they differ only in frequency and wavelength.

Diagram — the electromagnetic spectrum

A horizontal bar divided into regions, from lowest frequency and longest wavelength on the left to highest frequency and shortest wavelength on the right: radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, and gamma rays. The visible-light slice is enlarged into its colours — red, orange, yellow, green, blue, violet. Red has the longest wavelength and lowest frequency (about 3.84 × 10¹⁴ Hz); violet has the shortest wavelength and highest frequency (about 7.69 × 10¹⁴ Hz). Full values are in the electromagnetic-spectrum table.

Interference and superposition

Interference happens when waves arrive at the same point at the same time.
Constructive interference: crest meets crest, giving a larger amplitude.
Destructive interference: crest meets trough, giving a smaller amplitude.
Superposition: where pulses overlap, add their displacements to find the result.
Beats: two waves of slightly different frequency interfere to make beats. The beat frequency equals the difference between the two frequencies.

Standing waves resonance added from core curriculum

A standing wave appears to stay in one place. It forms when an incident wave and its reflection overlap in a confined region (a special case of interference).
Nodes are the stationary points; antinodes are the points of largest amplitude.
The fundamental is the lowest-frequency standing wave that fits in the region. Wavelength is measured node-to-node-to-node.
Resonance occurs when energy is transferred to a system at its natural frequency, building a large-amplitude standing wave.

Identical boundary conditions at both ends (such as a string fixed at both ends, or a pipe open at both ends):

\[ f_1 = \frac{v}{2L} \]

Fundamental (both ends alike) the fundamental frequency equals the wave speed divided by two times the length. All whole-number multiples of the fundamental can exist as harmonics.

Different boundary conditions at each end (such as a pipe closed at one end and open at the other):

\[ f_1 = \frac{v}{4L} \]

Fundamental (ends different) the fundamental frequency equals the wave speed divided by four times the length. Only the odd multiples of the fundamental can exist as harmonics.

The Doppler effect

As a wave source approaches, an observer meets waves more often, so the observed frequency is higher.
As a source moves away, the observed frequency is lower.
The source’s actual frequency does not change — only the apparent frequency does. The Doppler effect is not related to amplitude.

These animations compare a stationary source with a moving one. (They loop; the written description below each carries the information.)

Animation — stationary source
A source that stays still emits circular wavefronts as evenly spaced concentric circles. Because it is not moving, observers on every side measure the same wavelength and frequency — there is no Doppler shift.

Animation — moving source
As the source moves, the wavefronts **bunch together ahead** of it and **spread apart behind** it. An observer the source is approaching meets crests more often and hears a **higher** frequency (shorter wavelength); an observer it is moving away from hears a **lower** frequency (longer wavelength). The source’s own frequency never changes.

Diffraction added from core curriculum

Diffraction is the spreading of waves as they pass an obstacle or go through an opening. How much the wave spreads depends on the wavelength compared with the size of the opening or obstacle: the closer they are in size, the more the wave spreads.

Diagram — diffraction through a gap

Straight (plane) wavefronts travel from the left toward a barrier with a small gap in it. After passing through the opening the waves are no longer straight — they bend and spread out in curved (circular) wavefronts on the far side. The narrower the gap is compared with the wavelength, the more the waves spread. The same spreading happens when waves pass the edge of an obstacle.

Sound

Pitch depends on the sound wave’s frequency.
Loudness and the energy carried depend on the wave’s amplitude.
The speed of sound in air is about 331 m/s at STP (and roughly 340 m/s at ordinary room temperature).
People hear sounds from tens up to thousands of hertz; most musical notes are in the hundreds of hertz.

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9. Light, reflection & refraction added from core curriculum

This whole topic is required by the New York State core curriculum but was not on the fact sheet, so it is developed fully here.

Reflection

When a wave bounces off a surface, the angle it arrives at equals the angle it leaves at. Both angles are measured from the normal — an imaginary line drawn perpendicular to the surface.

\[ \theta_i = \theta_r \]

Law of reflection the angle of incidence equals the angle of reflection.

Diagram — reflection ray diagram

A ray (the incident ray) strikes a flat mirror at a point. A dashed normal line is drawn perpendicular to the mirror at that point. The angle of incidence (θ_i), measured between the incident ray and the normal, equals the angle of reflection (θ_r), measured between the normal and the reflected ray. Both angles are always measured from the normal, not from the mirror surface.

Refraction — why light bends

When a wave passes from one material into another, it can refract (bend) because its speed changes.
Entering a slower (optically denser, higher-index) material, light bends toward the normal. Entering a faster material, it bends away from the normal.
Crossing into a new material, the frequency stays the same, while the speed and wavelength change.

Index of refraction

The absolute index of refraction (n) compares the speed of light in a vacuum with its speed in a material. A larger n means light travels slower in that material and bends more. Index of refraction is inversely proportional to wave speed.

\[ n = \frac{c}{v} \]

Index of refraction the absolute index of refraction equals the speed of light in a vacuum divided by the speed of light in the material.

Snell’s law

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]

Snell’s law the index of refraction of the first medium times the sine of the angle in the first medium equals the index of refraction of the second medium times the sine of the angle in the second medium.

Diagram — refraction ray diagram (Snell’s law)

An incident ray travels through medium 1 (lower index n₁, e.g. air) and strikes the boundary with a denser medium 2 (higher index n₂, e.g. glass). The dashed normal is perpendicular to the boundary. Because the light slows down, the refracted ray bends toward the normal, so the angle of refraction θ₂ is smaller than the angle of incidence θ₁ (the faint dashed line shows the un-bent path for comparison). The angles obey Snell’s law, n₁ sin θ₁ = n₂ sin θ₂, with both angles measured from the normal.

The indices, speeds, and wavelengths in two media are related by:

\[ \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2} \]

Two-media relationship the ratio of the second index to the first equals the ratio of the first speed to the second speed, which also equals the ratio of the first wavelength to the second wavelength.

n: absolute index of refraction (no units)
c: speed of light in a vacuum, 3.00 × 10⁸ m/s
v: speed of light in the material, in meters per second
θ: angle measured from the normal
λ: wavelength, in meters

Common index values (air 1.00, water 1.33, crown glass 1.52, diamond 2.42, and more) are in the indices-of-refraction table.

Total internal reflection & the critical angle

When light tries to pass from a slower (higher-index) material into a faster (lower-index) one — for example from water or glass into air — it bends away from the normal. Past a certain incidence angle, called the critical angle, the light can no longer escape and instead reflects entirely back inside. This is total internal reflection, and it is how fiber-optic cables trap light.

Dispersion

A material’s index of refraction is slightly different for different colors (frequencies) of light, so white light separates into a spectrum when it refracts. This spreading by color is dispersion — it is what forms a rainbow and what a prism does. Violet light bends the most, red the least.

The electromagnetic spectrum

Visible light is a small part of the electromagnetic spectrum. All electromagnetic waves travel at the same speed in a vacuum (c). Ordered from lowest frequency (longest wavelength) to highest frequency (shortest wavelength): radio waves, microwaves, infrared, visible light, ultraviolet, X-rays, gamma rays.

Within visible light, from lowest to highest frequency the colors run red, orange, yellow, green, blue, violet. So red has the longest wavelength and lowest frequency; violet has the shortest wavelength and highest frequency. Full values are in the electromagnetic-spectrum table.

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10. Electricity & circuits

Electric charge

Charge is measured in coulombs (C).
Charge is conserved — it cannot be created or destroyed, though positive charge can neutralize negative charge.
The smallest unit of charge is the elementary charge, e = 1.60 × 10⁻¹⁹ C. Any measured charge is a whole-number multiple of e; a result that is not a multiple of e should be rejected.
One coulomb equals 6.25 × 10¹⁸ elementary charges. Lab charges are usually microcoulombs or nanocoulombs.

Coulomb’s law (force between charges)

\[ F_e = \frac{k q_1 q_2}{r^2} \]

Coulomb’s law the electrostatic force equals Coulomb’s constant times the product of the two charges, divided by the distance between their centers squared.

F_e: electrostatic force, in newtons
k: electrostatic (Coulomb’s) constant, 8.99 × 10⁹ N·m²/C²
q₁, q₂: the two charges, in coulombs
r: distance between the centers of the charges, in meters

Like gravitation, the electric force follows an inverse-square law. Unlike gravity, electric (and magnetic) forces can be either attractive or repulsive.

Electric field added from core curriculum

An electric field describes the force a charge would feel at a point in space. Its strength is found with a small positive test charge; the field points the way a positive charge would be pushed. The field between two oppositely charged parallel plates is uniform.

\[ E = \frac{F_e}{q} \]

Electric field strength electric field strength equals the electrostatic force divided by the charge.

Field lines point the way a positive test charge would be pushed: away from positive charges and toward negative charges. Where they are closer together the field is stronger.

Positive point charge

Field lines point straight out, away from the charge, evenly in all directions.

Negative point charge

Field lines point straight in, toward the charge, from all directions.

Two positive charges

Lines point away from both charges; between them they oppose, leaving a neutral point where the field is zero. Like charges repel.

Two negative charges

Lines point toward both charges; between them is a neutral point where the field cancels. Like charges repel.

Positive and negative (a dipole)

Field lines run from the positive charge to the negative charge, curving through the space between them. Opposite charges attract.

Charged parallel plates

Between two oppositely charged plates the field is uniform — evenly spaced, parallel lines pointing from the positive plate to the negative plate.

Voltage, current, and resistance

Voltage (potential difference) is provided by a battery and measured in volts. Rigorously, voltage is energy per charge.
Current is the flow of charge through a conductor, measured in amps. Rigorously, current is charge per time.
Resistance opposes current and is measured in ohms (Ω). It is provided by resistors, lamps, and other devices.
Power is energy per time.

\[ V = \frac{W}{q} \]

Potential difference voltage equals work (electrical energy) divided by charge.

\[ I = \frac{\Delta q}{t} \]

Current current equals the amount of charge that flows divided by the time.

\[ V = IR \qquad\Longleftrightarrow\qquad R = \frac{V}{I} \]

Ohm’s law voltage equals current times resistance; equivalently, resistance equals voltage divided by current.

At constant temperature, common metal conductors obey Ohm’s law.

Resistivity — what makes resistance

The resistance of a wire depends on its material and shape: longer and thinner wires have more resistance.

\[ R = \frac{\rho L}{A} \]

Resistance of a wire resistance equals the resistivity times the length of the conductor, divided by its cross-sectional area.

ρ: resistivity, a property of the material, in ohm-meters
L: length of the conductor, in meters
A: cross-sectional area, in square meters

Material resistivities are in the resistivities table.

Electrical power and energy

\[ P = VI = I^2 R = \frac{V^2}{R} \]

Electrical power power equals voltage times current, which equals current squared times resistance, which equals voltage squared divided by resistance.

\[ W = Pt = VIt = I^2Rt = \frac{V^2 t}{R} \]

Electrical energy electrical energy equals power times time, which equals voltage times current times time, which equals current squared times resistance times time, which equals voltage squared times time divided by resistance.

The brightness of a bulb depends on the power it dissipates.

Series circuits

Components in a series circuit lie in a single path. The same current flows through every part.

\[ I = I_1 = I_2 = I_3 = \dots \]\[ V = V_1 + V_2 + V_3 + \dots \]\[ R_{eq} = R_1 + R_2 + R_3 + \dots \]

Series rules in series, the total current equals the current through each resistor; the battery voltage equals the sum of the voltages across the resistors; and the equivalent resistance equals the sum of the individual resistances.

The sum of voltage changes around any loop is zero (Kirchhoff’s loop rule), which is conservation of energy.

Parallel circuits

In a parallel circuit the path for current divides and then rejoins. Every branch has the same voltage, and the equivalent resistance is less than any single branch.

\[ I = I_1 + I_2 + I_3 + \dots \]\[ V = V_1 = V_2 = V_3 = \dots \]\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \]

Parallel rules in parallel, the total current equals the sum of the branch currents; the voltage is the same across every branch; and one divided by the equivalent resistance equals the sum of one divided by each resistance.

Current into a junction equals current out of it (Kirchhoff’s junction rule), which is conservation of charge.

Kirchhoff’s rules in action

These animations illustrate the two rules. (They loop continuously; the written description below each one carries all the information.)

Junction rule (1st law) — conservation of charge

Animation — current at a T-junction
Current flows in along one wire and splits to flow out along the other two. The total current arriving at the junction equals the total current leaving it.

Animation — current at a four-way junction
Currents flow in and out along four wires. However the current divides, the amount entering the junction always equals the amount leaving — charge cannot pile up.

Loop rule (2nd law) — conservation of energy

Animation — voltages around one loop
A single-loop circuit with a battery and two resistors, labelled with voltages. The battery raises the potential by **+4 V**; the two resistors drop it by **1 V** and **3 V**. Around the loop the rises and drops add to zero: +4 − 1 − 3 = 0.

Animation — more than one loop
A battery with two resistors in parallel, showing that a circuit can contain several closed loops. The loop rule applies to *each* loop: around any loop, the voltage rises equal the voltage drops.

Meters

An ammeter measures current and is connected in series with the element.
A voltmeter measures voltage and is connected in parallel with the element.

Circuit symbols (cell, battery, switch, voltmeter, ammeter, resistor, variable resistor, lamp) are described in the circuit-symbols table.

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11. Magnetism & electromagnetism added from core curriculum

Moving electric charges produce magnetic fields. An electric current in a wire creates a magnetic field around it.
The magnetic field of a permanent magnet points out of the north (north-seeking) pole and into the south (south-seeking) pole outside the magnet.
Electromagnetic induction: relative motion between a conductor and a magnetic field can produce a potential difference (voltage) in the conductor. This is how generators work; the link between electricity and magnetism also runs motors.
Magnetic forces, like electric forces, can be attractive or repulsive — unlike gravity, which is only attractive. (Like poles repel; opposite poles attract.)
Energy can be stored in electric and magnetic fields and carried through conductors or space.

Diagram — field of a single bar magnet

A bar magnet with its north pole (N) on the left and south pole (S) on the right. Outside the magnet, curved field lines leave the north pole, loop around above and below, and return into the south pole, forming closed loops. The lines are most crowded — the field is strongest — at the poles.

Diagram — two bar magnets, opposite poles facing

Two bar magnets placed end to end with opposite poles facing: the north pole of the left magnet faces the south pole of the right magnet. Field lines run straight across the gap, from the N pole into the S pole, linking the two magnets — opposite poles attract. (If like poles faced each other, the field lines would bend sharply apart and the magnets would repel.)

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12. Modern physics added from core curriculum

This strand is required by the core curriculum and appears on the reference tables, but the fact sheet only touched it. It is developed fully here.

Quantization

On the atomic scale, energy and matter come in discrete amounts — they are quantized, not continuous.
Charge is quantized too. At the atomic level charge comes in multiples of the elementary charge (the charge on an electron or proton). At the subnuclear level, quarks carry fractions of the elementary charge.

The dual nature of light

Light behaves as both a wave (it diffracts and interferes) and a stream of particles called photons. On the atomic level both matter and energy show wave and particle characteristics. This wave–particle duality is the foundation of quantum mechanics.

Photons — the energy of light

Energy is emitted and absorbed in discrete packets called photons. A photon’s energy is proportional to its frequency (so higher-frequency light, like ultraviolet, carries more energy per photon than lower-frequency light, like infrared).

\[ E_{\text{photon}} = hf = \frac{hc}{\lambda} \]

Photon energy the energy of a photon equals Planck’s constant times the frequency, which also equals Planck’s constant times the speed of light divided by the wavelength.

E_photon: energy of the photon, in joules (or electronvolts)
h: Planck’s constant, 6.63 × 10⁻³⁴ J·s
f: frequency, in hertz
c: speed of light in a vacuum, 3.00 × 10⁸ m/s
λ: wavelength, in meters

The electronvolt (eV) is a small energy unit used for atoms: 1 eV = 1.60 × 10⁻¹⁹ J.

Atomic energy levels & spectra

Electrons in an atom can only have certain discrete energies, called energy levels. The lowest is the ground state; the level where the electron is freed from the atom is ionization (taken as 0 eV).
When an electron drops from a higher level to a lower one, the atom emits a photon whose energy is exactly the difference between the two levels. Absorbing a photon does the reverse. This is why each element has its own pattern of spectral lines.

\[ E_{\text{photon}} = E_i - E_f \]

Energy of a transition the energy of the emitted or absorbed photon equals the initial energy level minus the final energy level.

The reference tables show these as ladder diagrams — horizontal rungs on an energy axis whose spacing gets closer together as they approach the 0 eV ionization line at the top:

Diagram — hydrogen energy levels

Horizontal rungs on an energy axis: the ground state n=1 sits at −13.60 eV at the bottom, up to ionization at n=∞ and 0.00 eV at the top. The rungs crowd together near the top — n=2 at −3.40 eV, n=3 at −1.51 eV, and n=4, 5, 6 packed between −0.85 and −0.38 eV.

Diagram — mercury energy levels

The ground state (level a) sits at −10.38 eV at the bottom, up to ionization (level j) at 0.00 eV at the top, with levels b through i bunched between about −5.74 and −1.56 eV. All ten values are listed in the table.

Exact values for every level are in the energy-levels table.

Interactive simulation

See how electrons jumping between energy levels produce an element’s spectral lines:

Energy Levels & Spectra simulation (opens in a new tab)

Mass–energy equivalence

Mass and energy are two forms of the same thing. The fundamental source of all the energy in the universe is the conversion of mass into energy.

\[ E = mc^2 \]

Mass–energy energy equals mass times the speed of light squared.

The universal mass unit (u) is used for atomic masses; 1 u is equivalent to 9.31 × 10² MeV (megaelectronvolts) of energy.

The Standard Model of particle physics

The Standard Model explains what matter is made of. It has evolved from earlier models of the atom and states that atomic particles are built from smaller subnuclear particles, that the nucleus is a collection of quarks (which appear as protons and neutrons), and that every particle has a matching antiparticle with the opposite charge.

Diagram — classification of matter

Matter divides into two big families: hadrons and leptons. Hadrons split again into baryons (made of three quarks — protons and neutrons are baryons) and mesons (made of one quark plus one antiquark). Leptons (such as the electron) are not made of quarks.

Quarks come in six types. Up, charm, and top each carry a charge of +2/3 of an elementary charge. Down, strange, and bottom each carry −1/3.
Leptons also come in six types. The electron, muon, and tau each carry −1 elementary charge. The three neutrinos (electron, muon, and tau neutrinos) carry zero charge.

The full quark and lepton charges are in the Standard Model table.

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13. Complete equation reference

Every equation from the New York State Reference Tables for Physical Setting/Physics, grouped by section. Each is shown visually and written in words.

Mechanics

\[ \bar v = \frac{d}{t} \]

Average velocityaverage velocity equals displacement divided by time.

\[ a = \frac{\Delta v}{t} \]

Accelerationacceleration equals the change in velocity divided by time.

\[ v_f = v_i + at \]

Final velocityfinal velocity equals initial velocity plus acceleration times time.

\[ d = v_i t + \tfrac{1}{2}at^2 \]

Displacementdisplacement equals initial velocity times time plus one-half acceleration times time squared.

\[ v_f^{\,2} = v_i^{\,2} + 2ad \]

Velocity–displacementfinal velocity squared equals initial velocity squared plus two times acceleration times displacement.

\[ A_y = A\sin\theta \qquad A_x = A\cos\theta \]

Vector componentsthe y-component of a vector equals the vector times the sine of the angle; the x-component equals the vector times the cosine of the angle.

\[ a = \frac{F_{\text{net}}}{m} \]

Newton’s 2nd lawacceleration equals net force divided by mass.

\[ F_f = \mu F_N \]

Frictionthe force of friction equals the coefficient of friction times the normal force.

\[ F_g = \frac{G m_1 m_2}{r^2} \]

Gravitationthe gravitational force equals the gravitational constant times the two masses, divided by the distance between their centers squared.

\[ g = \frac{F_g}{m} \]

Gravitational fieldthe gravitational field strength equals weight divided by mass.

\[ p = mv \]

Momentummomentum equals mass times velocity.

\[ p_{\text{before}} = p_{\text{after}} \]

Momentum conservationtotal momentum before equals total momentum after.

\[ J = F_{\text{net}}\,t = \Delta p \]

Impulseimpulse equals net force times time, which equals the change in momentum.

\[ F_s = kx \]

Spring forcethe force on a spring equals the spring constant times the change in length from equilibrium.

\[ PE_s = \tfrac{1}{2}kx^2 \]

Spring energyspring potential energy equals one-half the spring constant times the change in length squared.

\[ F_c = m a_c \]

Centripetal forcecentripetal force equals mass times centripetal acceleration.

\[ a_c = \frac{v^2}{r} \]

Centripetal accelerationcentripetal acceleration equals speed squared divided by the radius.

\[ \Delta PE = mg\,\Delta h \]

Gravitational PEchange in gravitational potential energy equals mass times g times change in height.

\[ KE = \tfrac{1}{2}mv^2 \]

Kinetic energykinetic energy equals one-half mass times speed squared.

\[ W = Fd = \Delta E_T \]

Workwork equals force times displacement, which equals the change in total energy.

\[ E_T = PE + KE + Q \]

Total energytotal energy equals potential energy plus kinetic energy plus internal energy.

\[ P = \frac{W}{t} = \frac{Fd}{t} = F\bar v \]

Powerpower equals work over time, which equals force times displacement over time, which equals force times average velocity.

Electricity

\[ F_e = \frac{k q_1 q_2}{r^2} \]

Coulomb’s lawelectrostatic force equals Coulomb’s constant times the two charges divided by the distance between their centers squared.

\[ E = \frac{F_e}{q} \]

Electric fieldelectric field strength equals electrostatic force divided by charge.

\[ V = \frac{W}{q} \]

Potential differencevoltage equals work divided by charge.

\[ I = \frac{\Delta q}{t} \]

Currentcurrent equals charge divided by time.

\[ R = \frac{V}{I} \]

Resistance (Ohm’s law)resistance equals voltage divided by current.

\[ R = \frac{\rho L}{A} \]

Resistance of a wireresistance equals resistivity times length divided by cross-sectional area.

\[ P = VI = I^2 R = \frac{V^2}{R} \]

Electrical powerpower equals voltage times current, which equals current squared times resistance, which equals voltage squared divided by resistance.

\[ W = Pt = VIt = I^2Rt = \frac{V^2 t}{R} \]

Electrical energyelectrical energy equals power times time, which equals voltage times current times time, which equals current squared times resistance times time, which equals voltage squared times time divided by resistance.

\[ R_{eq} = R_1 + R_2 + R_3 + \dots \]

Series resistancein series, equivalent resistance equals the sum of the individual resistances.

\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \dots \]

Parallel resistancein parallel, one over the equivalent resistance equals the sum of one over each resistance.

Waves

\[ v = f\lambda \]

Wave speedwave speed equals frequency times wavelength.

\[ T = \frac{1}{f} \]

Periodthe period equals one divided by the frequency.

\[ \theta_i = \theta_r \]

Reflectionthe angle of incidence equals the angle of reflection.

\[ n = \frac{c}{v} \]

Index of refractionthe absolute index of refraction equals the speed of light in a vacuum divided by the speed in the material.

\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]

Snell’s lawthe first index times the sine of the first angle equals the second index times the sine of the second angle.

\[ \frac{n_2}{n_1} = \frac{v_1}{v_2} = \frac{\lambda_1}{\lambda_2} \]

Two-media relationshipthe ratio of the indices equals the inverse ratio of the speeds, which equals the ratio of the wavelengths.

Modern physics

\[ E_{\text{photon}} = hf = \frac{hc}{\lambda} \]

Photon energyphoton energy equals Planck’s constant times frequency, which equals Planck’s constant times the speed of light divided by wavelength.

\[ E_{\text{photon}} = E_i - E_f \]

Energy transitionphoton energy equals the initial energy level minus the final energy level.

\[ E = mc^2 \]

Mass–energyenergy equals mass times the speed of light squared.

Geometry & trigonometry

\[ A = bh \quad\text{(rectangle)} \qquad A = \tfrac{1}{2}bh \quad\text{(triangle)} \]

Areasarea of a rectangle equals base times height; area of a triangle equals one-half base times height.

\[ A = \pi r^2 \qquad C = 2\pi r \]

Circlearea of a circle equals pi r squared; circumference equals two pi r.

\[ c^2 = a^2 + b^2 \]

Pythagorean theoremthe hypotenuse squared equals the sum of the squares of the legs.

\[ \sin\theta = \frac{a}{c} \quad \cos\theta = \frac{b}{c} \quad \tan\theta = \frac{a}{b} \]

Right-triangle trigsine equals opposite over hypotenuse; cosine equals adjacent over hypotenuse; tangent equals opposite over adjacent.

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14. Reference data tables

The constants, values, and data from the reference tables, as accessible tables with row and column headers.

List of physical constants

Physical constants used in Regents Physics
Name	Symbol	Value
Universal gravitational constant	G	6.67 × 10⁻¹¹ N·m²/kg²
Acceleration due to gravity	g	9.81 m/s²
Speed of light in a vacuum	c	3.00 × 10⁸ m/s
Speed of sound in air at STP		3.31 × 10² m/s
Mass of Earth		5.98 × 10²⁴ kg
Mass of the Moon		7.35 × 10²² kg
Mean radius of Earth		6.37 × 10⁶ m
Mean radius of the Moon		1.74 × 10⁶ m
Mean distance — Earth to the Moon		3.84 × 10⁸ m
Mean distance — Earth to the Sun		1.50 × 10¹¹ m
Electrostatic constant	k	8.99 × 10⁹ N·m²/C²
1 elementary charge	e	1.60 × 10⁻¹⁹ C
1 coulomb	C	6.25 × 10¹⁸ elementary charges
1 electronvolt	eV	1.60 × 10⁻¹⁹ J
Planck’s constant	h	6.63 × 10⁻³⁴ J·s
1 universal mass unit	u	9.31 × 10² MeV
Rest mass of the electron	m_e	9.11 × 10⁻³¹ kg
Rest mass of the proton	m_p	1.67 × 10⁻²⁷ kg
Rest mass of the neutron	m_n	1.67 × 10⁻²⁷ kg

Prefixes for powers of 10

Metric prefixes
Prefix	Symbol	Notation
tera	T	10¹²
giga	G	10⁹
mega	M	10⁶
kilo	k	10³
deci	d	10⁻¹
centi	c	10⁻²
milli	m	10⁻³
micro	µ	10⁻⁶
nano	n	10⁻⁹
pico	p	10⁻¹²

Approximate coefficients of friction

Approximate coefficients of friction (kinetic and static)
Surfaces	Kinetic	Static
Rubber on concrete (dry)	0.68	0.90
Rubber on concrete (wet)	0.58	—
Rubber on asphalt (dry)	0.67	0.85
Rubber on asphalt (wet)	0.53	—
Rubber on ice	0.15	—
Waxed ski on snow	0.05	0.14
Wood on wood	0.30	0.42
Steel on steel	0.57	0.74
Copper on steel	0.36	0.53
Teflon on Teflon	0.04	—

Absolute indices of refraction

Measured with light of frequency 5.09 × 10¹⁴ Hz.

Absolute indices of refraction
Material	Index (n)
Air	1.00
Corn oil	1.47
Diamond	2.42
Ethyl alcohol	1.36
Glass, crown	1.52
Glass, flint	1.66
Glycerol	1.47
Lucite	1.50
Quartz, fused	1.46
Sodium chloride	1.54
Water	1.33
Zircon	1.92

Resistivities at 20°C

Resistivities of materials at 20 degrees Celsius
Material	Resistivity (Ω·m)
Aluminum	2.82 × 10⁻⁸
Copper	1.72 × 10⁻⁸
Gold	2.44 × 10⁻⁸
Nichrome	150. × 10⁻⁸
Silver	1.59 × 10⁻⁸
Tungsten	5.60 × 10⁻⁸

The electromagnetic spectrum

Regions in order of increasing frequency (decreasing wavelength). All travel at 3.00 × 10⁸ m/s in a vacuum.

Electromagnetic spectrum regions, lowest to highest frequency
Region	Relative frequency & wavelength
Long radio waves	Lowest frequency, longest wavelength
Radio waves (AM, FM, TV)	Low frequency, long wavelength
Microwaves	↑ frequency increases, ↓ wavelength decreases
Infrared	Just below visible
Visible light	About 3.84 × 10¹⁴ Hz (red) to 7.69 × 10¹⁴ Hz (violet)
Ultraviolet	Just above visible
X-rays	High frequency, short wavelength
Gamma rays	Highest frequency, shortest wavelength

Approximate frequencies of visible-light colors
Color	Frequency (Hz)
Red (longest wavelength)	3.84 × 10¹⁴
Orange	4.82 × 10¹⁴
Yellow	5.03 × 10¹⁴
Green	5.20 × 10¹⁴
Blue	6.10 × 10¹⁴
Violet (shortest wavelength)	6.59–7.69 × 10¹⁴

Energy-level diagrams

Energy levels for the hydrogen atom
Level (n)	Energy (eV)
n = 1 (ground state)	−13.60
n = 2	−3.40
n = 3	−1.51
n = 4	−0.85
n = 5	−0.54
n = 6	−0.38
n = ∞ (ionization)	0.00

A few energy levels for the mercury atom
Level	Energy (eV)
a (ground state)	−10.38
b	−5.74
c	−5.52
d	−4.95
e	−3.71
f	−2.68
g	−2.48
h	−1.57
i	−1.56
j (ionization)	0.00

Particles of the Standard Model

Charges are given as multiples of the elementary charge, e. For every particle there is a matching antiparticle with the opposite charge.

Quarks
Name	Symbol	Charge
Up	u	+2/3 e
Charm	c	+2/3 e
Top	t	+2/3 e
Down	d	−1/3 e
Strange	s	−1/3 e
Bottom	b	−1/3 e

Leptons
Name	Symbol	Charge
Electron	e	−1 e
Muon	μ	−1 e
Tau	τ	−1 e
Electron neutrino	ν_e	0
Muon neutrino	ν_μ	0
Tau neutrino	ν_τ	0

Classification of matter

Matter → hadrons and leptons. Hadrons → baryons (three quarks; protons and neutrons) and mesons (one quark + one antiquark).

Circuit symbols

Standard circuit symbols and what they represent
Component	How the symbol is drawn
Cell	One long thin line (positive) and one short thick line (negative), side by side
Battery	Two or more cells in a row
Switch	A line with a break and a hinged segment that opens or closes the gap
Voltmeter	A circle with the letter V, connected in parallel
Ammeter	A circle with the letter A, connected in series
Resistor	A zig-zag line (or, internationally, a rectangle)
Variable resistor	A resistor with a diagonal arrow drawn across it
Lamp	A circle with an X inside

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How to use this guide

1. Measurement, units & math tools

Vectors and scalars

Components of any vector A at angle θ from the horizontal

The metric system & SI units

Geometry & trigonometry you need

2. Kinematics (motion)

Definitions

Position–time graphs

Velocity–time graphs

Acceleration

Special equations for displacement

Algebraic kinematics — the method

The three kinematics equations

Projectile motion

3. Forces & Newton’s laws (dynamics)

Equilibrium

Newton’s first law (inertia)

Newton’s second law

Solving force problems — the two-step process

A free-body diagram includes

Mass and weight

Normal force

Components of a diagonal force

Inclined planes (ramps)

Friction

Newton’s third law

Two-body problems

Hooke’s law (springs)

4. Circular motion & gravitation

Uniform circular motion

Universal gravitation

5. Work, energy & power

Forms of energy

Work

The work–energy theorem — the method

Power

Energy in collisions

6. Momentum & impulse

Momentum

Impulse

Conservation of momentum

7. What is conserved?

8. Waves & sound

Wave anatomy

Key facts

Relating frequency, period, wavelength, and speed

Transverse and longitudinal waves

The electromagnetic spectrum

Interference and superposition

Standing waves resonance added from core curriculum

The Doppler effect

Diffraction added from core curriculum

Sound

9. Light, reflection & refraction added from core curriculum

Reflection

Refraction — why light bends

Index of refraction

Snell’s law

Total internal reflection & the critical angle

Dispersion

The electromagnetic spectrum

10. Electricity & circuits

Electric charge

Coulomb’s law (force between charges)

Electric field added from core curriculum

Voltage, current, and resistance

Resistivity — what makes resistance

Electrical power and energy

Series circuits

Parallel circuits

Kirchhoff’s rules in action

Junction rule (1st law) — conservation of charge

Loop rule (2nd law) — conservation of energy

Meters

11. Magnetism & electromagnetism added from core curriculum

12. Modern physics added from core curriculum

Quantization

The dual nature of light

Photons — the energy of light