SUVAT Equations Explained, with Worked Examples

October 6, 2026

The SUVAT equations describe motion in a straight line with constant acceleration. The five letters are s (displacement), u (initial velocity), v (final velocity), a (acceleration) and t (time). Each equation links four of the five, so if you know any three you can find the other two.

The five SUVAT equations

Units: s in metres (m), u and v in metres per second (m/s), a in metres per second squared (m/s²), t in seconds (s).

SUVAT at GCSE vs A-Level

At GCSE Physics, the one you use most is v² − u² = 2as (the same as v² = u² + 2as), alongside acceleration = change in velocity ÷ time and distance = speed × time. On AQA it appears on the physics equation sheet, so you need to be able to use it rather than recall it. Check your own board's sheet.

At A-Level (Maths mechanics and Physics), you need all five, often in two-stage problems, with vertical motion under gravity and with negative values for direction. A-Level Maths expects you to know them; you may also be asked to derive them from a velocity–time graph.

Worked examples

Example 1: finding final velocity and distance

A car travelling at 4 m/s accelerates at 2 m/s² for 5 s. Find its final speed and the distance travelled.

s = ?, u = 4, v = ?, a = 2, t = 5.
v = u + at = 4 + 2 × 5 = 14 m/s.
s = ut + ½at² = 4 × 5 + ½ × 2 × 5² = 20 + 25 = 45 m.

Example 2: an object dropped from rest (GCSE)

A ball is dropped from a height of 20 m. Ignoring air resistance, how fast is it moving when it hits the ground? (g = 9.8 m/s²)

u = 0 (dropped from rest), a = 9.8, s = 20, v = ?
v² = u² + 2as = 0 + 2 × 9.8 × 20 = 392, so v = √392 = 19.8 m/s (3 s.f.).

Example 3: finding deceleration

A train slows from 30 m/s to rest over a distance of 450 m. Find its deceleration and how long it takes to stop.

u = 30, v = 0, s = 450, a = ?
v² = u² + 2as gives 0 = 900 + 900a, so a = −1 m/s² (a deceleration of 1 m/s²).
v = u + at gives 0 = 30 − t, so t = 30 s.

Example 4: projected vertically upwards (A-Level)

A ball is thrown straight up at 14 m/s. Find its maximum height and the total time until it returns to the point of projection. (g = 9.8 m/s²)

Take upwards as positive, so a = −9.8. At the top, v = 0.
v² = u² + 2as gives 0 = 196 − 19.6s, so s = 10 m.
For the whole flight s = 0: s = ut + ½at² gives 0 = 14t − 4.9t² = t(14 − 4.9t), so t = 14 ÷ 4.9 = 2.86 s (3 s.f.).

Example 5: a quadratic in t (A-Level)

A particle starts with velocity 3 m/s and accelerates uniformly at 2 m/s². How long does it take to travel 40 m?

u = 3, a = 2, s = 40, t = ?
s = ut + ½at² gives 40 = 3t + t², so t² + 3t − 40 = 0 and (t + 8)(t − 5) = 0.
Time can't be negative, so t = 5 s.

Common mistakes that cost marks

• Using SUVAT when acceleration isn't constant. The equations only work for constant (uniform) acceleration. If acceleration changes, use graphs (or calculus at A-Level).
• Forgetting direction. Choose a positive direction and stick to it. Gravity is −9.8 m/s² if up is positive.
• Mixing up "from rest" and "comes to rest". "From rest" means u = 0; "comes to rest" means v = 0.
• Wrong units. Convert km/h to m/s (divide by 3.6) and minutes to seconds before substituting.
• Square-rooting too early. In v² = u² + 2as, find v² first, then take the square root at the end.

How the equations link to velocity–time graphs

On a velocity–time graph, the gradient is the acceleration and the area under the line is the displacement. For constant acceleration the graph is a straight line from u to v, so the area is a trapezium: s = ½(u + v)t. The gradient gives a = (v − u) ÷ t, which rearranges to v = u + at. The other three equations come from combining these two.

For the full topic, including motion graphs and projectiles, read our A-Level kinematics and SUVAT revision notes, then test yourself with auto-marked quizzes.

Frequently asked questions

What does SUVAT stand for?

SUVAT stands for the five quantities in the equations: s (displacement), u (initial velocity), v (final velocity), a (acceleration) and t (time).

When can you use the SUVAT equations?

Only for motion in a straight line with constant (uniform) acceleration, such as free fall when air resistance is ignored.

Which SUVAT equation do I use?

List s, u, v, a and t, fill in the three you know, and choose the equation that does not contain the fifth quantity, the one you neither know nor need.

Are the SUVAT equations given in the GCSE exam?

At GCSE, v² − u² = 2as appears on the AQA physics equation sheet, so you need to use it but not memorise it. At A-Level Maths you are expected to know all five.