SUVAT equations, also known as the kinematic equations

The SUVAT equations, called the kinematic equations in the US, are five equations for motion in a straight line with constant acceleration. They link displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t), and each one leaves out a different letter. Know any three of the letters and you can find the other two.

This SUVAT calculator finds the other two values from any three. It starts on a car doing 8 m/s that accelerates at 2 m/s² for 5 s, the same numbers as example 1 below. For more room, open the full SUVAT calculator.

What does SUVAT stand for?

SUVAT is the five letters in the equations, one for each quantity. US textbooks write some of them differently, but they mean the same things.

All 5 SUVAT equations and kinematic equations

These are the 5 SUVAT equations, also called the SUVAT formulas or formulae. They are the same as the kinematic equations for constant acceleration. Each one uses four of the five letters, and the letter it leaves out is how you choose it.

They work only when two things are true: the acceleration is constant, and the motion is along a straight line. A car braking steadily fits. So does a ball in free fall, if you ignore air resistance. A car whose acceleration keeps changing does not, and needs a graph or calculus instead.

Why are there 3, 4 or 5 kinematic equations?

The physics is the same everywhere. The only difference is how many equations a course chooses to print.

• 3 kinematic equations: the AP Physics 1 reference sheet lists v = v₀ + at, Δx = v₀t + ½at² and v² = v₀² + 2aΔx, written with x − x₀ for Δx.
• 4 kinematic equations: the usual US list, often called the big 4, adds Δx = ½(v₀ + v)t. UK A-level Physics data sheets print the same four.
• 5 kinematic equations: the UK SUVAT list adds s = vt − ½at², the one without u. A-level Maths formula booklets print all five.

No list is missing anything. Every equation can be built from v = u + at and s = ½(u + v)t, as the derivation further down shows, so learn the list your course uses.

Finding each letter: SUVAT without t, a, u or v

Write s, u, v, a and t in a column and fill in what the question gives you. "From rest" means u = 0, and "stops" means v = 0. Mark the letter you want. One letter is now left over, the one you neither know nor want, so choose the equation that leaves it out.

Try one now. Choose the equation for this problem, then press Mark to check your answer in the course.

Rearranging for each letter

Rearrange before you put the numbers in. These are the forms you will use most.

SUVAT equations for t

Two equations give t directly, as the table shows. The other two contain t², so they are quadratics in t. The quadratic formula turns them into:

A quadratic can give two answers. Keep the one that makes sense, as example 4 shows.

Worked examples

To learn this from the start, the Laws of Motion course builds up to these equations through velocity-time graphs, then moves on to projectile motion.

Example 1: how far and how fast

A car travelling at 8 m/s accelerates at 2 m/s² for 5 s. How far does it travel, and how fast is it going at the end?

You know u = 8, a = 2 and t = 5. You want s and v. You do not have v yet, so find s with the equation that leaves out v:

Then find v with the equation that leaves out s:

The car travels 65 m and ends at 18 m/s. The calculator at the top of this page gives the same answers.

Example 2: braking with no time given

A car doing 72 km/h brakes steadily and stops in 50 m. Find its deceleration and the time it takes.

Convert first: 72 ÷ 3.6 = 20 m/s. So u = 20, v = 0 and s = 50. Time is not given or wanted yet, so use SUVAT without t:

The minus sign means it is slowing down, so the deceleration is 4 m/s². For the time, s = ½(u + v)t is the simplest choice:

The car stops in 5 s.

Vertical motion under gravity

Near Earth, anything moving freely up or down has a constant acceleration of about 9.8 m/s² downwards if you ignore air resistance, so SUVAT applies. Choose one direction as positive and keep it for the whole problem. If up is positive, a = −9.8 m/s² the entire time, on the way up as well as on the way down. At the highest point v = 0, but a is still −9.8 m/s².

Example 3: a ball thrown straight up

A ball is thrown straight up at 19.6 m/s. Find its greatest height and how long it takes to come back to the start.

Take up as positive, so u = 19.6 and a = −9.8. At the top v = 0, and no time is needed, so use SUVAT without t:

The greatest height is 19.6 m. Back at the start the displacement is zero, so s = 0 in the equation without v:

So t = 0, which is the moment of the throw, or t = 19.6 ÷ 4.9 = 4 s.

Example 4: a ball thrown up from a bridge

A ball is thrown straight up at 15 m/s from a bridge 20 m above a river. How long does it take to land in the water?

Up is positive and the river is below the start, so s = −20, u = 15 and a = −9.8. You do not need v, so use s = ut + ½at² and tidy it into a quadratic:

This gives t = 4.07 or t = −1.00. A negative time is before the ball was thrown, so reject it. The ball takes 4.07 s.

Here x is time in seconds and y is height above the river in metres. The curve crosses zero twice, and those are the two answers. Only the one after the throw is real.

Two-stage problems

If the acceleration changes part way through, the journey as a whole does not have one constant acceleration. Split it at the change and work on each stage separately. The final velocity of stage 1 is the initial velocity of stage 2.

Example 5: a model rocket

A model rocket starts from rest and accelerates upwards at 15 m/s² for 4 s. The engine then cuts out and it moves freely under gravity. Find its greatest height, ignoring air resistance.

Stage 1, engine on: u = 0, a = 15 and t = 4.

Stage 2, engine off: the rocket is still rising, so u = 60, v = 0 at the top and a = −9.8.

The greatest height is 120 + 183.7 = 304 m to 3 significant figures.

Here x is time and y is velocity. Stage 2 lasts 60 ÷ 9.8 = 6.12 s, so the rocket reaches the top after 10.12 s. The area under a velocity-time graph is the displacement, and this triangle has area ½ × 10.12 × 60 = 304 m, which matches. A quick sketch like this is a good way to check a multi-stage answer.

Kinematic equations for projectile motion

A projectile follows a curve, so you cannot use SUVAT on the whole path at once. Split it into horizontal and vertical motion instead. Without air resistance the horizontal acceleration is zero and the vertical acceleration is g. The only thing the two directions share is the time.

Say a ball is kicked horizontally at 15 m/s off a 20 m cliff. Taking down as positive, u = 0 vertically, a = 9.8 and s = 20:

Horizontally the speed stays at 15 m/s, so the ball lands 15 × 2.02 = 30.3 m from the cliff. You can launch your own in the projectile motion simulator.

Deriving the SUVAT equations

Every SUVAT equation comes from a velocity-time graph, and the kinematic equations are derived the same way. In the UK, A-level Maths expects you to derive them as well as use them.

This is the car from example 1, with x as time and y as velocity. Constant acceleration makes a straight line from u at the start to v at time t, and the outline shows the area under it.

1. The gradient (slope) is the acceleration. Rearranging gives the first equation.

2. The area under the line is the displacement. The shape is a trapezium (trapezoid) with parallel sides u and v and width t.

3. Replace v to get s = ut + ½at². Put v = u + at into the area equation. You can see this on the graph too: a rectangle ut plus a triangle ½at². For the car, that is 40 + 25 = 65 m.

4. Replace u to get s = vt − ½at². Put u = v − at into the area equation.

5. Replace t to get v² = u² + 2as. Put t = (v − u)/a into the area equation, then multiply both sides by 2a.

Common mistakes

• Using SUVAT when the acceleration changes. If it changes part way, split the motion into stages. If it changes all the time, use a graph or calculus.
• Missing a hidden value. "From rest" and "dropped" mean u = 0. "Stops" and "at the highest point" mean v = 0.
• Mixing up signs. Pick a positive direction before you start. Deceleration means a is negative, and anything against your positive direction is negative too.
• Treating s as total distance. s is the displacement from the start. A ball that lands back in your hand has s = 0, however far it travelled.
• Ignoring the negative square root. v² = 49 gives v = 7 or v = −7. Decide which direction fits the question.
• Mixing units. Change km/h to m/s by dividing by 3.6, and minutes to seconds, before you start.
• Rounding too early. Keep the full calculator value between stages and round only at the end.

Are the SUVAT equations in the formula booklet?

Some of them, depending on the exam. This table was checked against the exam boards' own sheets in November 2026.

So A-level Maths gives you all five and A-level Physics gives you four. GCSE Physics gives you only v² − u² = 2as and the equation for acceleration, so practise those two. Whatever the exam, you still have to choose the right equation and rearrange it yourself.

In England, Ofqual requires GCSE Physics and Combined Science equation sheets for the life of the current specifications, and AQA has published its sheet for June 2027. Sheets can change, so check the one for your exam series.

Stuck on one? Put your three values into the SUVAT calculator to check your answer, then work back through the method.

Practise SUVAT in a short timed paper

A guide shows you the method. A timed paper shows whether it has stuck. A-level Maths: Mechanics is the closest match, because constant acceleration comes up in nearly every paper. A-level Physics: Year 1 covers more topics, so SUVAT will not be in every paper. In the US, the same practice opens as AP Physics 1 and AP Physics C: Mechanics. Each short paper has 8 questions worth 25 to 35 marks, and there is a new one every day.

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Physics · MechanicsSUVAT calculator: enter any 3 values, get the other 2Enter any three of s, u, v, a and t to find the other two, see which SUVAT equation was used and why, and get both answers when there are two.