SUVAT equations, explained
The SUVAT equations describe motion with constant acceleration. They appear in A level Maths mechanics and in physics. Here is how to use them with confidence.
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What SUVAT stands for
- s: displacement, in metres
- u: initial velocity, in m/s
- v: final velocity, in m/s
- a: acceleration, in m/s²
- t: time, in seconds
The five equations
- v = u + at (no s)
- s = ut + ½at² (no v)
- s = vt − ½at² (no u)
- v² = u² + 2as (no t)
- s = ½(u + v)t (no a)
Each equation leaves out exactly one of the five quantities. That is the trick to choosing.
How to choose the right equation
- Write the five letters in a column.
- Fill in the three you know.
- Mark the one you want.
- The remaining letter is the one you do not need, so pick the equation that leaves it out.
Worked example
A car starts from rest and accelerates at 2 m/s² for 5 seconds. How far does it travel?
- u = 0, a = 2, t = 5, s = ? (v not needed)
- Use s = ut + ½at² = 0 + ½ × 2 × 25 = 25 m
Worked example: vertical motion
A ball is thrown straight up at 14 m/s. How high does it go? Take up as positive and g = 9.8 m/s².
- u = 14, v = 0 at the top, a = −9.8, s = ? (t not needed)
- Use v² = u² + 2as: 0 = 196 − 19.6s, so s = 10 m
Common mistakes
- Using SUVAT when acceleration is not constant.
- Forgetting that displacement and velocity have directions. Choose a positive direction and keep it.
- Using g = 9.8 when the question tells you to use a different value.
- Mixing up displacement and total distance travelled.
Where SUVAT appears
SUVAT is part of A level Maths mechanics and A level Physics. The equation v² = u² + 2as also appears in GCSE Physics. Check your board's formula booklet: on many papers you are expected to know these equations rather than be given them.
Where the equations come from
You do not have to take the five equations on trust. They all come from one velocity-time graph.
- The gradient of the line is the acceleration: a = (v − u)/t. Rearranged, that is v = u + at.
- The area under the line is a trapezium with parallel sides u and v and width t, so s = ½(u + v)t.
- Substitute v = u + at into s = ½(u + v)t and you get s = ut + ½at².
- Substitute u = v − at instead and you get s = vt − ½at².
- Eliminate t between v = u + at and s = ½(u + v)t and you get v² = u² + 2as.
Knowing this means you can rebuild any equation you forget in the exam, and it explains why every one of them needs constant acceleration: a straight line on the graph.
More worked examples
Example 3: braking to a stop
A car travelling at 20 m/s brakes uniformly and stops in 50 m. Find the deceleration and the time taken.
- Known: u = 20, v = 0, s = 50. Unknown: a, then t.
- v² = u² + 2as gives 0 = 400 + 100a, so a = −4 m/s² (a deceleration of 4 m/s²).
- v = u + at gives 0 = 20 − 4t, so t = 5 s.
Example 4: dropping an object
A stone is dropped from rest from a height of 45 m. Take g = 9.8 m/s² and downwards as positive.
- u = 0, a = 9.8, s = 45.
- s = ut + ½at² gives 45 = 4.9t², so t² ≈ 9.18 and t ≈ 3.03 s.
- v² = u² + 2as gives v² = 2 × 9.8 × 45 = 882, so v ≈ 29.7 m/s.
Example 5: two stages of motion
A cyclist starts from rest, accelerates at 3 m/s² for 4 s, then rides at a constant speed for 6 s. How far do they travel?
- Stage 1: v = u + at = 0 + 3 × 4 = 12 m/s, and s = ½at² = ½ × 3 × 16 = 24 m.
- Stage 2: constant speed, so s = 12 × 6 = 72 m. SUVAT is not needed when a = 0.
- Total distance: 96 m.
Split multi-stage problems at every change of acceleration. The final velocity of one stage is the initial velocity of the next.
Example 6: a horizontal projectile
A ball is kicked horizontally at 15 m/s from the top of a 20 m cliff. How far from the base of the cliff does it land? Take g = 9.8 m/s².
- Vertically: u = 0, a = 9.8, s = 20. So 20 = 4.9t² and t ≈ 2.02 s.
- Horizontally: constant velocity, so distance = 15 × 2.02 ≈ 30.3 m.
The two directions are solved separately and linked only by the time.
A checklist for every SUVAT question
- Is the acceleration constant? If not, use calculus or graphs instead.
- Choose a positive direction and write it down.
- List s, u, v, a and t, filling in the three you know.
- Pick the equation without the quantity you neither know nor need.
- Check the sign and the units of your answer, and whether it is sensible.
Practice questions
Try each one before opening the answer.
Q1. A car accelerates uniformly from 5 m/s to 25 m/s in 8 s. Find its acceleration and the distance travelled.
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Q2. A stone is thrown straight up at 20 m/s. Find its maximum height and the time to reach it. Use g = 9.8 m/s².
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Q3. A train decelerates uniformly from 30 m/s to rest over 450 m. Find the deceleration and the time taken.
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Q4. A particle moves in a straight line with u = 4 m/s and a = −2 m/s². When does it return to its starting point?
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Frequently asked questions
Can I use SUVAT for a car going round a bend?
Not for the whole motion. SUVAT works in one straight line at a time. For projectiles, apply it separately to the horizontal and vertical directions.
What value of g should I use?
Use whatever the question gives. If it does not say, 9.8 m/s² is standard in A level Maths, and some physics papers use 9.81 m/s².
What if acceleration changes with time?
Then SUVAT does not apply. In A level Maths you integrate and differentiate instead: v = ∫a dt and s = ∫v dt.