Equations of motion: v = u + at

Beginner

Final velocity under constant acceleration.

Key relationv = u + at

For constant acceleration a, velocity grows linearly with time: v = u + at, where u is the initial velocity. It comes from integrating a = dv/dt. Use it when you know u, a, t and want v.

Worked example

A bike at 2 m/s accelerating 0.5 m/s² for 8 s reaches 6 m/s.

Common mistake

It only works for constant acceleration.

In more detail

For constant acceleration, the velocity after time t is v = u + at, where u is the initial velocity and a is the acceleration. It is the basic relation between velocity, acceleration and time.

How to solve it

Write down the known quantities: initial velocity u, acceleration a and time t. Note their signs using one chosen positive direction.
Check that the acceleration is constant over the whole interval. The equation does not apply otherwise.
Substitute into v = u + at and solve for the unknown.
Attach units. Velocity should come out in metres per second if you used SI units throughout.

Worked examples

Example 1: A car starts from rest and accelerates uniformly at 2 m/s² for 5 s. What is its final speed?

u = 0, a = 2 m/s², t = 5 s. So v = 0 + (2)(5) = 10 m/s, or 36 km/h.

Example 2: A train travelling at 20 m/s brakes at 1.5 m/s². What is its speed after 8 s?

Take the direction of motion as positive, so a = −1.5 m/s². Then v = 20 + (−1.5)(8) = 20 − 12 = 8 m/s. The negative sign shows that the velocity is falling, not that the train is reversing.

Checking your answer

Velocity change should equal acceleration times time. If the acceleration is 2 m/s², each second should add 2 m/s to the speed, so 5 seconds adds 10 m/s.

Real-world uses

Estimating the final speed of a car accelerating from rest for a given time.
Calculating the speed of a falling object after a set time.
A drag racer starts from rest and accelerates steadily, and the final speed tells the driver how fast the car is going at the finish line.

Open the interactive lesson, with simulation and quiz →

Frequently asked questions

When can I use this equation?

Only when acceleration is constant, such as in uniform acceleration problems.

What if the initial velocity is zero?

Then v = at, which is the simplest case.

Which equation should I use when time is not given?

Use v² = u² + 2as, because it relates velocities and displacement without needing time.