Acceleration & the Kinematic Equations

You’ll be able to

Define acceleration as the rate of change of velocity
Select the correct kinematic equation from the known and unknown quantities
Solve constant-acceleration problems, including braking to a stop

Acceleration is how fast velocity changes

Acceleration is the change in velocity per unit time, a = Δv / Δt, measured in m/s². It is a vector. Speeding up, slowing down, and turning are all accelerations. Crucially, a negative acceleration does not mean "slowing down" — it means acceleration points in the negative direction. An object slows down only when its acceleration points opposite to its velocity.

Constant acceleration unlocks three equations

When acceleration is constant, the whole motion is captured by a small set of kinematic equations. Each links four of the five quantities — initial velocity v₀, final velocity v, acceleration a, displacement Δx, and time t — leaving one out. You choose an equation by asking which variable you don’t have and don’t need.

Velocity–time (no Δx)

v = v₀ + at

Use when you have or want time but not displacement.

Position–time (no v)

Δx = v₀t + ½at²

The ½ is not optional — dropping it is the single most common kinematics mistake.

Timeless equation (no t)

v² = v₀² + 2aΔx

The rescue equation when time is neither given nor asked for.

Worked example

A car starts from rest and accelerates at 3 m/s² for 5 s. Find its final velocity and how far it travels.

1.List knowns: v₀ = 0, a = 3 m/s², t = 5 s.
2.Final velocity uses v = v₀ + at = 0 + (3 m/s²)(5 s) = 15 m/s.
3.Displacement uses Δx = v₀t + ½at² = 0 + ½(3 m/s²)(5 s)².
4.Evaluate: ½ × 3 × 25 = 37.5 m.

Answer: v = 15 m/s and Δx = 37.5 m

Tip

Before solving, write down the five symbols v₀, v, a, Δx, t and fill in what you know. The equation to use is the one containing your three knowns plus the single unknown you want.

Checkpoint

A cart moving at 10 m/s accelerates at 2 m/s² for 4 s. What is its final velocity?

Checkpoint

A car traveling at 20 m/s brakes at a constant −4 m/s² until it stops. How far does it travel while stopping?

Checkpoint

A ball is dropped from rest and falls for 3 s with a = 10 m/s². How far does it fall?

On the exam

When a problem gives you velocities and a distance but never mentions time, reach straight for v² = v₀² + 2aΔx. Recognizing that missing variable saves you from solving a needless quadratic.

Answer the 3 checkpoints as you read.