1. Kinematics

Key Kinematic Equations

v = v₀ + at

x = v₀t + ½at²

v² = v₀² + 2ax

x = ½(v₀ + v)t

Δx = vt (constant speed)

1.1 Kinematic Definitions

Displacement (Δx or Δy): How far an object ends up from its initial position, regardless of total distance traveled.

Average velocity (v̄): Displacement divided by time interval: v̄ = Δx/Δt

Instantaneous velocity (v): How fast an object is moving at a specific moment in time.

Average Speed: Total distance divided by time duration.

1.2 Position-time Graphs

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

Four Basic Position-time Graph Shapes for Uniform Acceleration

Getting Slower

Negative Acceleration

Getting Faster

Positive Acceleration

Positive then Negative

Acceleration Changes

Negative then Positive

Acceleration Changes

1.3 Velocity-time Graphs

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

Important: On velocity-time graphs, you cannot determine the object's location.

1.4 Acceleration

Acceleration: How much an object's speed changes in one second.

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

1.7 Projectile Motion

Projectile Motion Key Points

Vertical acceleration: g ≈ 10 m/s² (downward)
Horizontal acceleration: 0 (no air resistance)
Horizontal motion: Δx = v₀t (constant velocity)
Independent motions: Horizontal and vertical components separate
Velocity addition: Use Pythagorean theorem: v = √(vₓ² + vᵧ²)

1.8 Rotational Kinematics

Angular displacement (θ): Angle through which object rotates (radians)

Angular velocity (ω): Rate of angular displacement (rad/s)

Angular acceleration (α): Rate of change of angular velocity (rad/s²)

Linear-Angular Relationships:

Linear displacement: x = rθ
Linear speed: v = rω
Linear acceleration: a = rα

2. Dynamics

Key Force Equations

F = ma

W = mg

f = μN

Fᶜ = mv²/r

2.1 Definition of Equilibrium

An object is in equilibrium if it is:

At rest, OR
Moving in a straight line at constant speed

Equilibrium condition: Net force equals zero (ΣF = 0)

2.2 Newton's Second Law

Newton's Second Law: F = ma

Net force is in the direction of acceleration
F and a are vectors pointing in same direction
Force causes acceleration, not velocity

2.3 Solving Force Problems

Problem-Solving Strategy:

Draw free-body diagram
Choose coordinate system
Apply Newton's second law in each direction
Solve the resulting equations

2.4 Mass vs Weight

Mass (m): Intrinsic property, measured in kg

Weight (W): Force due to gravity, W = mg, measured in N

Mass and weight are different! Weight depends on gravitational field strength.

2.5-2.10 Types of Forces

Force Components

When angle θ is measured from horizontal: Fx = F cos θ, Fy = F sin θ

Normal Force (N):

Contact force perpendicular to surface
Does NOT always equal weight
Equals perpendicular component of all forces

Friction Force (f):

f = μN (μₛ for static, μₖ for kinetic)
Opposes motion, acts parallel to surface
Static friction ≥ kinetic friction (μₛ > μₖ)

Inclined Plane Components:

Parallel to incline: mg sin θ
Perpendicular to incline: mg cos θ

Inclined Plane Force Components

Weight components on inclined plane: parallel = mg sin θ, perpendicular = mg cos θ

2.11 Newton's Third Law

For every action, there is an equal and opposite reaction.

Action-reaction pairs act on DIFFERENT objects
Equal in magnitude, opposite in direction
Occur simultaneously

3. Circular Motion

Circular Motion Equations

aᶜ = v²/r = ω²r

Fᶜ = maᶜ = mv²/r

T = 2π/ω

f = 1/T

Centripetal acceleration (aᶜ): Acceleration directed toward center of circular path

Centripetal force (Fᶜ): Net force toward center providing centripetal acceleration

Centripetal force is NOT a separate force - it's the NET force toward center
Examples: tension, friction, gravity, normal force can provide centripetal force
At constant speed, acceleration still exists (direction changes)

4. Conservation Laws

Conservation principles are fundamental laws of physics that state certain quantities remain constant in isolated systems.

4.1 Conservation of Energy

In the absence of non-conservative forces, total mechanical energy remains constant:
E = K + U = constant

4.2 Conservation of Momentum

In the absence of external forces, total momentum of a system remains constant:
Σpᵢ = Σpf = constant

4.3 Conservation of Angular Momentum

In the absence of external torques, total angular momentum remains constant:
L = Iω = constant

5. Energy

Energy Equations

K = ½mv²

U = mgh

W = F·d = Fd cos θ

P = W/t = F·v

5.1 Kinetic Energy (K)

Energy of motion: K = ½mv²

5.2 Potential Energy (U)

Gravitational Potential Energy:

U = mgh (near Earth's surface)
Reference point is arbitrary (often ground level)
Only changes in potential energy matter physically

Elastic Potential Energy:

U = ½kx² (for springs)
x is displacement from equilibrium
k is spring constant

5.3 Work (W)

Work is energy transferred to or from an object: W = F·d cos θ

Work is positive when force and displacement are in same direction
Work is negative when force opposes displacement
Only component of force parallel to displacement does work

5.4 Power (P)

Rate of energy transfer: P = W/t = F·v

6. Momentum

Momentum Equations

p = mv

J = Δp = F·Δt

F = Δp/Δt

6.1 Momentum (p)

Momentum is mass times velocity: p = mv (vector quantity)

6.2 Impulse (J)

Impulse equals change in momentum: J = Δp = F·Δt

Impulse-Momentum Theorem: The impulse on an object equals its change in momentum

6.3 Collisions

Elastic Collisions:

Both momentum AND kinetic energy are conserved
Objects bounce apart
Relative speed before = relative speed after

Inelastic Collisions:

Only momentum is conserved
Kinetic energy decreases (converted to other forms)
Objects may stick together (completely inelastic)

7. Rotation

Rotational Equations

τ = rF sin θ = r⊥F

I = Σmr²

K = ½Iω²

L = Iω

7.1 Torque (τ)

Torque is rotational force: τ = rF sin θ = r⊥F

r⊥ is perpendicular distance from rotation axis to line of force
Positive torque causes counterclockwise rotation
Net torque causes angular acceleration: τ = Iα

7.2 Moment of Inertia (I)

Rotational inertia: resistance to angular acceleration

Common Moments of Inertia:

Point mass: I = mr²
Rod (center): I = (1/12)mL²
Rod (end): I = (1/3)mL²
Disk: I = (1/2)mr²
Sphere: I = (2/5)mr²

7.3 Rotational Kinetic Energy

K = ½Iω²

7.4 Angular Momentum (L)

L = Iω (conserved when no external torques act)

8. Simple Harmonic Motion

SHM Equations

F = -kx

T = 2π√(m/k)

T = 2π√(L/g)

x = A cos(ωt + φ)

8.1 Spring Force

Hooke's Law: F = -kx

k is spring constant (N/m)
x is displacement from equilibrium
Negative sign indicates restoring force

8.2 Mass-Spring System

Period: T = 2π√(m/k)

Period is independent of amplitude!

8.3 Simple Pendulum

Period: T = 2π√(L/g) (for small angles)

Period is independent of mass and amplitude (small angles)
L is length of pendulum
Only valid for angles < 15°

8.4 Energy in SHM

Total energy: E = ½kA² (constant)
At equilibrium: all kinetic energy
At maximum displacement: all potential energy
Energy oscillates between kinetic and potential

9. Fluids

Fluid Equations

P = P₀ + ρgh

Fᵦ = ρfluid·Vdisplaced·g

A₁v₁ = A₂v₂

ρ = m/V

9.1 Pressure

Pressure increases with depth: P = P₀ + ρgh

P₀ is atmospheric pressure
ρ is fluid density
h is depth below surface

9.2 Buoyant Force

Archimedes' Principle: Fᵦ = ρfluid·Vdisplaced·g

Buoyant force equals weight of displaced fluid
Object floats if ρobject < ρfluid
Object sinks if ρobject > ρfluid

9.3 Fluid Flow

Continuity Equation: A₁v₁ = A₂v₂

Flow rate is constant in a pipe (conservation of mass)

9.4 Bernoulli's Principle

As fluid speed increases, pressure decreases

10. Graph Analysis

10.1 Motion Graphs

Position-Time Graphs:

Slope = velocity
Curved line = changing velocity (acceleration)
Horizontal line = at rest
Straight line = constant velocity

Velocity-Time Graphs:

Slope = acceleration
Area under curve = displacement
Horizontal line = constant velocity
Straight line = constant acceleration

10.2 Force Graphs

Force-Time Graphs:

Area under curve = impulse
Impulse = change in momentum

10.3 Energy Graphs

Energy-Position Graphs:

Total energy line is horizontal (if conservative)
Turning points where K = 0
Equilibrium where dU/dx = 0

10.4 Five Types of Mathematical Relationships

Constant

y = b

No relationship

Linear

y = mx + b

Direct proportional

Inverse

y = m(1/x) + b

Graph y vs 1/x

Quadratic

y = mx² + b

Graph y vs x²

Square Root

y² = mx + b

Graph y² vs x

Note: There are no exponential relationships in AP Physics 1

Quick Reference - Key Equations

Kinematics

v = v₀ + at
x = x₀ + v₀t + ½at²
v² = v₀² + 2a(x - x₀)
x = x₀ + ½(v₀ + v)t

Forces & Motion

F = ma
W = mg
f = μN
Fᶜ = mv²/r
τ = rF sin θ

Energy & Work

K = ½mv²
U = mgh
U = ½kx²
W = Fd cos θ
P = W/t

Momentum & Impulse

p = mv
J = Δp = FΔt
F = Δp/Δt

Rotation

K = ½Iω²
L = Iω
v = rω
a = rα

Waves & SHM

F = -kx
T = 2π√(m/k)
T = 2π√(L/g)
v = fλ

AP Physics 1 Fact Sheet