Equations of motion: v² = u² + 2as

Beginner

Relates velocities and displacement without time.

Key relationv² = u² + 2as

Eliminating time between the two earlier equations gives v² = u² + 2as. It is the fastest route when time is unknown or irrelevant, and it is the kinematic form of the work–energy theorem.

Worked example

Braking from 20 m/s at −5 m/s²: 0 = 400 − 10s → s = 40 m.

Common mistake

Taking the square root gives ±; choose the sign by physical reasoning.

In more detail

The equation v² = u² + 2as links velocities and displacement without needing time. It is useful when the time is unknown.

How to solve it

Identify u, v, a and s. Time is not needed for this equation.
Make sure the acceleration is constant over the distance considered.
Rearrange to s = (v² − u²)/(2a) if you want the distance.
Take the square root only at the end, and choose the sign that makes physical sense.

Worked examples

Example 1: A car speeds up from 0 to 25 m/s at a constant 2.5 m/s². How far does it travel?

s = (v² − u²)/(2a) = (625 − 0)/(2 × 2.5) = 625/5 = 125 m.

Example 2: A car travelling at 30 m/s brakes at 6 m/s² until it stops. What distance does it cover?

Here v = 0, u = 30 and a = −6 m/s². So s = (0 − 900)/(2 × −6) = −900/−12 = 75 m.

Checking your answer

Doubling the speed should roughly quadruple the stopping distance, since the equation depends on u squared. Going from 30 m/s to 60 m/s at the same braking should give 300 m instead of 75 m.

Real-world uses

Finding the braking distance needed to stop a car from a given speed.
Estimating the length of runway needed for an aircraft to reach take-off speed.
Engineers estimate the braking distance of a car from its initial speed and its deceleration without needing to measure the time taken.

Open the interactive lesson, with simulation and quiz →

Frequently asked questions

How do I choose which equation to use?

List the known quantities and pick the equation that contains the unknown, but leaves out the quantity you do not know.

Why do signs matter in these equations?

Because displacement, velocity and acceleration are vectors, so their signs show direction along the chosen axis.

What if the object is slowing down?

Use a negative value for a. The formula then gives the correct smaller final velocity or the distance covered before stopping.