Kinematics Equations Cheat Sheet

A printable reference covering displacement, velocity, acceleration, constant-acceleration equations, free fall, and motion graphs for grades 9-11.

Download PNG

Kinematics describes motion using position, displacement, velocity, acceleration, and time. This cheat sheet helps students choose the right equation, identify known and unknown quantities, and keep vector directions consistent. It is especially useful for one-dimensional motion, free fall, and interpreting motion graphs. Students need these formulas because many physics problems use the same core relationships in different situations. The main ideas are based on changes in position and velocity over time. Average velocity is $v_{avg} = \frac{\Delta x}{\Delta t}$ , and average acceleration is $a_{avg} = \frac{\Delta v}{\Delta t}$ . For constant acceleration, the equations $v = v_0 + at$ , $\Delta x = v_0t + \frac{1}{2}at^2$ , and $v^2 = v_0^2 + 2a\Delta x$ connect motion quantities. Motion graphs show the same relationships visually, where slope and area often give important physical meaning.

Key Facts

Displacement is change in position, so $\Delta x = x_f - x_i$ and direction matters.
Average velocity is $v_{avg} = \frac{\Delta x}{\Delta t}$ , while average speed is $\frac{\text{total distance}}{\Delta t}$ .
Average acceleration is $a_{avg} = \frac{\Delta v}{\Delta t} = \frac{v - v_0}{t}$ .
For constant acceleration, final velocity is found with $v = v_0 + at$ .
For constant acceleration, displacement is found with $\Delta x = v_0t + \frac{1}{2}at^2$ .
When time is not known, use $v^2 = v_0^2 + 2a\Delta x$ .
For free fall near Earth, acceleration is usually $a = -9.8\ \text{m/s}^2$ if upward is chosen as positive.
On a velocity-time graph, the slope equals acceleration and the area under the graph equals displacement.

Vocabulary

Position: Position is an object's location relative to a chosen origin, often written as $x$ .
Displacement: Displacement is the change in position, written $\Delta x = x_f - x_i$ , and includes direction.
Velocity: Velocity is the rate of change of position, so average velocity is $v_{avg} = \frac{\Delta x}{\Delta t}$ .
Acceleration: Acceleration is the rate of change of velocity, written $a = \frac{\Delta v}{\Delta t}$ for average acceleration.
Free Fall: Free fall is motion under the influence of gravity alone, with $a = g \approx 9.8\ \text{m/s}^2$ downward near Earth.
Constant Acceleration: Constant acceleration means the value of $a$ does not change, allowing the standard kinematics equations to be used.

Common Mistakes to Avoid

Confusing distance with displacement is wrong because distance has no direction, while displacement can be positive, negative, or zero.
Using the wrong sign for acceleration is wrong because the direction of $a$ must match the chosen coordinate system, such as $a = -9.8\ \text{m/s}^2$ when upward is positive.
Mixing up initial and final velocity is wrong because $v_0$ is the velocity at the start and $v$ is the velocity at the end of the time interval.
Using constant-acceleration equations when acceleration changes is wrong because formulas like $v = v_0 + at$ assume $a$ stays constant.
Forgetting units is wrong because quantities such as $\text{m}$ , $\text{s}$ , $\text{m/s}$ , and $\text{m/s}^2$ identify what the number physically means.

Practice Questions

1 A cyclist starts from rest and accelerates at $2.5\ \text{m/s}^2$ for $6.0\ \text{s}$ . Find the final velocity using $v = v_0 + at$ .
2 A car moving at $20\ \text{m/s}$ slows down at $-4.0\ \text{m/s}^2$ for $3.0\ \text{s}$ . Find its displacement using $\Delta x = v_0t + \frac{1}{2}at^2$ .
3 A ball is dropped from rest off a building and falls for $2.0\ \text{s}$ . Using $a = 9.8\ \text{m/s}^2$ downward, find the distance it falls.
4 A velocity-time graph is a horizontal line above the time axis. Explain what this shows about the object's velocity, acceleration, and displacement.