Physics Unit 1: Kinematics Cheat Sheet

Quick-reference study sheet

5 Steps to Solving a Physics Problem

1Draw a picture interpreting the word problem
2List your knowns and your unknowns
3Select the appropriate equation for the problem
4Solve the equation symbolically first
5Check your answer — does it make sense?

1.1 Scalars and Vectors in One Dimension

Scalar

Vector

Magnitude only

e.g. distance, speed, time, mass

Magnitude AND direction

e.g. displacement, velocity, acceleration, force

Sign Convention as Direction (1-D)

Pick a positive direction (usually right or up). Anything in that direction is positive; the opposite direction is negative. The sign of a vector quantity is its direction.

Example: Vector Addition/Subtraction in 1-D

A car moves +12 m, then moves −5 m (backs up). Find total displacement.

Add the signed values: 12 + (−5)
Answer: +7 m (7 m in the positive direction)

⚠ Watch out:In 1-D, "subtracting" a vector means adding its negative — flip the sign, then add.

1.2 Displacement, Velocity, Speed and Acceleration

Distance	Displacement
Scalar; total path length traveled. Always positive.	Vector; straight-line change in position, Δx = x_f − x_i. Can be positive, negative, or zero.

Average Speed

speed = total distancetotal time

Scalar — uses total distance traveled, not displacement, so it is always ≥ the magnitude of average velocity.

Average Velocity

v_avg = ΔxΔt

Displacement over the total time interval

Instantaneous Velocity

v = slope of x-t graph at a point

Velocity at one specific moment in time

Average Acceleration

a_avg = ΔvΔt

Change in velocity over the total time interval

Instantaneous Acceleration

a = slope of v-t graph at a point

Acceleration at one specific moment in time

Sign of Acceleration Relative to Velocity

Same sign (both + or both −): object speeding up

Opposite signs: object slowing down

a = 0: constant velocity

v = 0 momentarily at rest

Common Units & Examples

Quantity	SI Unit	Other Common Units	Typical Example
Distance / Displacement	meter (m)	km, cm, ft, mi	Length of a football field ≈ 91 m
Time	second (s)	min, hr	A typical class period ≈ 50 min
Velocity / Speed	m/s	km/h, mph	Highway speed limit 65 mph ≈ 29 m/s
Acceleration	m/s²	km/h/s, g's	Free fall ≈ 9.8 m/s² (1 g)

1.3 Representing Motion

Graph	Slope Gives	Area Under Curve Gives
Position–time (x-t)	Velocity	—
Velocity–time (v-t)	Acceleration	Displacement
Acceleration–time (a-t)	—	Change in velocity

A quantity described in words, shown on a graph, and written as an equation are three views of the same motion — practice translating freely among them (e.g. "constant positive velocity" → straight diagonal line on x-t graph → x = x₀ + vt).

Position vs. Time Graphs

Slope	Velocity	Acceleration	Motion
Slope constant, zero	Zero velocity	Zero	Stopped
Slope constant, positive	Positive velocity, constant	Zero	Constant forward velocity
Slope positive, increasing	Positive velocity, increasing	Positive	Increasing forward velocity
Slope positive, decreasing	Positive velocity, decreasing	Negative	Decreasing forward velocity

Velocity vs. Time Graphs

Slope	Velocity	Acceleration	Motion
Slope is zero	Constant	Acceleration is zero	At rest, or moving at constant velocity
Slope is positive	Increasing from negative to positive	Acceleration is constant, positive	Moving in the negative direction and slowing down, then speeding up in the positive direction
Slope is negative	Decreasing from positive to negative	Acceleration is constant, negative	Moving in the positive direction and slowing down, then speeding up in the negative direction (e.g. a ball tossed straight up)
Slope positive, decreasing	Increasing, leveling off toward a maximum	Positive, decreasing toward zero	Speeding up quickly at first, then leveling off near a maximum speed (e.g. a skydiver approaching terminal velocity)
Slope negative, approaching zero	Decreasing, leveling off toward zero	Negative, decreasing in magnitude toward zero	Slowing down quickly at first, then leveling off as it approaches rest (e.g. friction bringing an object to a stop)

Examples of How Motion Looks with 3 Graphs

Motion	Position–Time Graph	Velocity–Time Graph	Acceleration–Time Graph
Stationary Object
Uniform Motion
Motion with Constant Acceleration

Motion Graphs

Written Description	Motion Diagram	Position vs. Time	Velocity vs. Time	Acceleration vs. Time
Positive direction, speeding up	T=0▶
Negative direction, speeding up	T=0◀
Positive direction, slowing down	T=0▶
Negative direction, slowing down	T=0◀
Turning around	T=0⇄

Kinematic Equations (constant acceleration)

The 5 Kinematic Formulas

a = ΔvΔt or v_f − v₀Δt

Δx = v₀t + 12at²

v_f² = v₀² + 2aΔx

v_f = v₀ + at

Δx = v_f + v₀2Δt

Variables

a = acceleration
Δv = change in velocity
v_f = velocity final
v₀ = velocity initial
Δt = change in time
Δx = displacement

Example: A car starts at rest and accelerates at 3.0 m/s² for 4.0 s. Find its displacement.

Given: v₀ = 0, a = 3.0 m/s², t = 4.0 s
Use Δx = v₀t + 12at² = 0 + 12(3.0)(4.0)²
Answer: Δx = 24 m

Free Fall

Motion under gravity alone (no air resistance). Use the kinematic equations with a = g = 9.8 m/s² (or -9.8 m/s² if up is positive). All objects fall with the same acceleration regardless of mass.

1.4 Reference Frames and Relative Motion

Inertial Reference Frame

An inertial reference frame is a viewpoint or coordinate system where an object with no outside forces acting on it stays still or moves in a straight line at a steady speed. In this special framework, Newton's first law of motion is fully true, and you do not need fake or imaginary forces to explain how things move

Relative Velocity

v_AC = v_AB + v_BC

Velocity of A relative to C = velocity of A relative to B + velocity of B relative to C (vector addition; works in 1-D and 2-D)

Example: A boat moves at 4 m/s east relative to the water. The river flows at 2 m/s east relative to the shore. Find the boat's velocity relative to the shore.

v_boat,shore = v_boat,water + v_water,shore
v_boat,shore = 4 m/s + 2 m/s
Answer: 6 m/s east

⚠ Watch out:Keep subscript order consistent and add velocities as vectors — in 2-D this means adding components, not magnitudes.

1.5 Vectors and Motion in Two Dimensions

Vector Components

v_x = v cosθ

v_y = v sinθ

θ measured from the positive x-axis. Magnitude: v = √v_x² + v_y². Perpendicular axes (x and y) are independent — motion in one direction does not affect motion in the other.

Projectile Motion — Horizontal Launch

x-direction: v_x = v₀ (constant), x = v₀t

y-direction: v_y = −gt, y = −12gt²

Initial vertical velocity is zero; time to land is set entirely by the height dropped.

Projectile Motion — Launched at an Angle

Time of Flight

t = 2v₀ sinθg

Max Height

H = v₀² sin²θ2g

Range

R = v₀² sin(2θ)g

Example: A ball is launched at 20 m/s at 30° above the horizontal. Find the time of flight.

Use t = 2v₀ sinθg
t = 2(20)sin(30°)9.8 = 209.8
Answer: t ≈ 2.0 s

⚠ Watch out:Range and max-height formulas assume launch and landing heights are equal. If they differ, break the motion into x and y components and solve with the kinematic equations directly instead of using the shortcut formulas.