Reading Position-time Graphs:

• Vertical axis: Distance from detector
• Slope steepness: Speed of object
• Curved graphs: Use tangent line slope for instantaneous speed

Direction from Slope:

• Front slash (/) slope: Moving away from detector
• Back slash (\) slope: Moving toward detector

Acceleration from Shape:

• Concave up (U): Positive acceleration
• Concave down (∩): Negative acceleration

Reading Velocity-time Graphs:

• Vertical axis: Speed of object
• Above horizontal axis: Moving away from detector
• Below horizontal axis: Moving toward detector
• Area under curve: Displacement
• Slope: Acceleration

Acceleration Direction:

• Speeding up: Acceleration in direction of motion
• Slowing down: Acceleration opposite to direction of motion
• Free fall: g ≈ 10 m/s² (near Earth's surface)
• Units: m/s² or m/s/s

• When an object is moving at a constant speed \(v\), its displacement is given by \(\Delta x = vt\)
  • This includes horizontal components of projectile motion
• When an object starts at rest and speeds up, or when an object slows to a stop, its displacement is given by either \(x = \frac{1}{2}at^2\) or \(x = v_0 t + \frac{1}{2}at^2\)

You must follow these steps to solve an algebraic kinematics calculation:

1. Define a positive direction, i.e. the direction "away from the detector." Label that direction.
2. Indicate in words what portion of motion you are considering, e.g. "motion from launch to the peak of the flight."
3. Fill out a cross diagram chart, including signs and units, of the five kinematics variables:
   • \(\Delta x\) [displacement]
   • \(v_0\) [initial velocity]
   • \(a\) [acceleration]
   • \(v_f\) [final velocity]
   • \(t\) [time for motion to happen]
4. If three of the five variables are known, the problem is solvable; use the kinematics equations to solve.

When an object is in free-fall:

• Its VERTICAL acceleration is always \(g\) or 10 m/s² near the surface of the earth.
• Its HORIZONTAL acceleration is always zero. Meaning the only equation you use for the horizontal motion is \(\Delta x = vt\)

• Velocities in perpendicular directions add with the Pythagorean theorem, just like perpendicular forces.
• The magnitude of an object's velocity is known as its speed.
• To approach a projectile problem, make two kinematics charts: one vertical, one horizontal.

• For all forces other than the force of the earth, objects must be in contact in order to experience a force.
• An object's acceleration is in the direction in which forces are unbalanced.
• The net force is in the direction in which the forces are unbalanced.
• The net force is in the direction of acceleration.

The two-step problem solving process:

1. Draw a free-body diagram.
   • 1a. Break angled forces into components, if necessary.
2. Write two equations, one for Newton's second law in each direction:
   • (up forces) − (down forces) = \(ma_y\)
   • (left forces) − (right forces) = \(ma_x\)

• Mass tells how much material is contained in an object.
• The units of mass are kilograms (kg).
• Weight is the force of a planet acting on an object (in newtons).
• On earth's surface, the gravitational field is 10 N/kg. This means that on earth, 1 kg of mass weighs 10 N.

• A normal force is the force of a surface on an object in contact with that surface.
• A normal force acts perpendicular to a surface.
• A platform scale (i.e. bathroom scale) reads the normal force.

• To determine the resultant force, use the Pythagorean theorem on the horizontal and vertical components.
• The "magnitude" of a force means the amount of the resultant force.

When the angle of the diagonal force is measured from the horizontal:

• The horizontal component of the force is the magnitude of the force times \(\cos \theta\) or \(F_x = F \cos \theta\)
• The vertical component of the force is the magnitude of the force times \(\sin \theta\) or \(F_y = F \sin \theta\)

1. A labeled arrow representing each force. Each arrow begins on the object and points in the direction in which the force acts.
2. A list of the forces, indicating the object applying the force and the object experiencing the force.

When an object is on a ramp, break the object's weight into components parallel to and perpendicular to the incline. Do not use x and y axes.

• The component of the object's weight parallel to the incline is \(mg \sin \theta\)
• The component of the object's weight perpendicular to the incline is \(mg \cos \theta\)

• The force of friction is the force of a surface on an object acting along the surface.
• The force of friction acts in the opposite direction of an object's motion (relative to a surface).
• The coefficient of friction is not a force.
• The coefficient of friction is a number that tells how sticky two surfaces are.
• The force of friction is equal to the coefficient of friction times the normal force: \(F_f = \mu F_N\)
• The coefficient of kinetic friction is used when an object is moving; the coefficient of static friction is used when an object is not moving. Both types of coefficients of friction obey the same equation.
• The coefficient of static friction can take on any value up to a maximum, which depends on the properties of the materials in contact.
• For two specific surfaces in contact, the maximum coefficient of static friction is greater than the coefficient of kinetic friction.

• Newton's Third Law says that the force of object A on object B is equal to the force of object B on object A.
• A "Third Law Force Pair" is a pair of forces that obeys Newton's third law.
• Two forces in a third law force pair can never act on the same object.

In a two-body problem, usually:

• Draw one free body diagram per object
• Write \(F_{net} = ma\) for each object separately.
• Acceleration is the same for each object
• One rope = one tension

• An object moving at constant speed \(v\) in a circle of radius \(r\) has an acceleration of magnitude \(\frac{v^2}{r}\), directed toward the center of the circle.
• Centripetal Force is the net force that creates uniform circular motion.
  • The centripetal force is the sum of all of the radial forces (forces along the radius)

• All massive objects attract each other with a gravitational force.
• The gravitational force \(F_G\) of one object on another is given by:

• The gravitational field \(g\) near an object of mass \(M\) is given by: \(g = \frac{GM}{d^2}\)
  • where \(d\) represents the distance from the object's center to anywhere you're considering.
• The weight of an object near a planet is given by \(mg\), where \(g\) is the gravitational field due to the planet at the object's location.
• The gravitational field near a planet is always equal to the free-fall acceleration.
• Gravitational mass is measured by measuring an object's weight using \(W = mg\)
• Inertial mass is measured by measuring the net force on an object, measuring the object's acceleration, and using \(F_{net} = ma\).
• In all experiments ever performed, gravitational mass is equal to inertial mass.

In a circular orbit of a satellite around a planet, consider the planet-satellite system:

• Kinetic energy is constant (same speed)
• Gravitational potential energy is constant (same orbital radius)
• Angular momentum \(mvr\) is constant (no external torques)
• Total mechanical energy is constant (no external work, and no internal energy)

In an elliptical orbit of a satellite around a planet, consider the planet-satellite system:

• Kinetic energy is NOT constant (speed changes)
• Gravitational potential energy is NOT constant (orbital radius changes)
• Angular momentum \(mvr\) is constant (no external torques)
• Total mechanical energy is constant (no external work, and no internal energy)

• Mechanical energy is conserved when there is no net work done by external forces. (And when there's no internal energy conversion.)
• Angular momentum is conserved when no net external torque acts.
• Momentum in a direction is conserved when no net external force acts in that direction.

All forms of energy have units of joules, abbreviated J.

• Kinetic energy: \(K = \frac{1}{2}mv^2\). Here, \(m\) is the mass of the object, and \(v\) is its speed.
• Gravitational potential energy: \(U_g = mgh\). Here, \(m\) is the mass of the object, \(g\) is the gravitational field strength, and \(h\) is the vertical height of the object above its lowest position.
• The term mechanical energy refers to the sum of a system's kinetic and potential energy.
• Spring potential energy: \(U_s = \frac{1}{2}kx^2\). Here, \(k\) is the spring constant, and \(x\) is the distance the spring is stretched or compressed from its equilibrium position.
• Rotational kinetic energy: \(K_r = \frac{1}{2}I\omega^2\). Here, \(I\) is the rotational inertia of the object, and \(\omega\) is the angular speed of the object.
• Internal energy is heat energy that causes an increase in the temperature of the system.

Problem-solving process:

• Define the object or system being described.
• Define the two positions you're considering.
• Draw an annotated energy bar chart.
• Write an equation based on the energy bar chart, with one term per bar:
  • \(E_{initial} \pm Work = E_{final}\)

• Positive work is done by a force parallel to an object's displacement.
• Negative work is done by a force antiparallel to an object's displacement.
• No work is done by a force acting perpendicular to an object's displacement.
• Work is a scalar quantity – it can be positive or negative, but does not have a direction.
• The area under a force vs. displacement graph is work.

• Power is defined as the amount of work done in one second, or energy used in one second: \(P = \frac{W}{t}\)
• The units of power are joules per second, which are also written as watts.
• An alternate way of calculating power when a constant force acts is \(P = F\bar{v}\)

• Near the surface of a planet, the potential energy of a planet-object system is \(mgh\), with \(h = 0\) at the lowest point of the motion.
• Away from the surface, the potential energy of a planet-object system is treated differently:
  • PE is larger the farther from the planet's center.
  • PE has a negative value (except when the object is way far away from the planet, in which case PE is zero).
  • The equation for potential energy is \(PE = -\frac{GMm}{r}\). Don't use this equation unless you must derive an expression. The negative sign is confusing.

• In an elastic collision, mechanical energy of the system is conserved.
• Collisions for which objects stick together can not be elastic.
• Collisions for which objects bounce off each other may or may not be elastic.

• Momentum is equal to mass times velocity: \(p = mv\)
• The standard units of momentum are newton·seconds, abbreviated N·s. (Equivalent to kg·m/s)
• The direction of an object's momentum is always the same as its direction of motion.

• Impulse, \(J\), can be calculated in either of two ways:
  1. Impulse is equal to the change in an object's momentum
  2. Impulse is equal to the force experienced multiplied by the time interval of collision, \(J = F\Delta t\)
• Impulse has the same units as momentum, N·s.
• Impulse is the area under a force vs. time graph.

• When no external forces act on a system of objects, the system's momentum can not change.
• The total momentum of two objects before a collision is equal to the objects' total momentum after the collision.
• Momentum is a vector: that is, total momentum of two objects moving in the same direction adds together; total momentum of two objects moving in opposite directions subtracts.
• A system's center of mass obeys Newton's second law: that is, the velocity of the center of mass only changes when an external net force acts on the system.
• The location of the center of mass is given by \(M_{tot}x_{cm} = m_1x_1 + m_2x_2 + ...\)

• Rotational inertia \(I\) represents an object's resistance to angular acceleration.
• For a point particle, rotational inertia is \(MR^2\), where \(M\) is the particle's mass, and \(R\) is the distance from the axis of rotation.
• Rotational inertia of multiple objects add together algebraically.

• Before calculating angular momentum, it is necessary to define a rotational axis.
• The angular momentum \(L\) of an object is given by:
  • \(I\omega\) for an extended object
  • \(mvr\) for a point object. For an object moving in a straight line at constant speed, \(r\) represents the "distance of closest approach."

• When no torques act external to a system, angular momentum of the system cannot change.
• Angular momentum is a vector – angular momentums in the same sense add, angular momentums in opposite senses subtract.
• Angular momentum is conserved separately from linear momentum. Do not combine them in a single equation.

• Many objects that vibrate back-and-forth exhibit simple harmonic motion. The pendulum and the mass-on-a-spring are the most common examples of simple harmonic motion.
• An object in simple harmonic motion makes a position-time graph in the shape of a sine function. The equation for position as a function of time is \(x = A \cos(2\pi ft)\).
• An object in simple harmonic motion experiences a net force whose:
  • magnitude increases as a linear function of distance from the equilibrium position
  • direction always points toward the equilibrium position

Definitions involving simple harmonic motion:

• Amplitude (A): The maximum distance from the equilibrium position reached by an object in simple harmonic motion
• Period (T): The time for an object to complete one entire vibration.
• Frequency (f): How many entire vibrations an object makes each second.

• A spring pulls with more force the farther the spring is stretched or compressed.
• The force of a spring is given by the equation \(F = kx\). Here, \(k\) is the spring constant of the spring, and \(x\) is the distance the spring is stretched or compressed from its equilibrium position.
• The spring constant is a property of a spring, and is always the same for the same spring.
• The standard units of the spring constant are N/m.

• When dealing with an object hanging vertically from a spring, it's easiest to consider the spring-earth-object system.
• The potential energy of the spring-earth-object system is \(PE = \frac{1}{2}kx^2\), where \(x\) is measured from the position where the object would hang in equilibrium.

• The pressure in a static column of fluid is \(P = P_0 + \rho gh\); Here the \(\rho gh\) term is called the "gauge pressure," meaning the pressure above atmospheric.
• Density, \(\rho\), is defined as mass/volume. Thus, mass can be expressed as \(\rho V\).
• The buoyant force on an object is equal to the weight of the fluid displaced.
• The equation for the buoyant force is \(F_B = \rho Vg\), where \(\rho\) is the density of the FLUID and \(V\) is the volume SUBMERGED.

• The continuity principle is a statement of conservation of mass: the volume flow rate (or mass flow rate) must be the same everywhere.
• The continuity principle for flow of cross sectional area \(A\) and speed \(v\) says \(A_1v_1 = A_2v_2\).
• Bernoulli's equation is a statement of conservation of energy.
• Bernoulli's equation says \(P + \rho gh + \frac{1}{2}\rho v^2\) is constant at any two locations.

The table below summarizes the five types of mathematical relationships you need to know for AP Physics 1: