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Equations of motion calculator

Solve the equations of motion for constant acceleration. Give any three of displacement, initial velocity, final velocity, acceleration and time, and get the other two with every step of the working.

Units: metres, seconds, m/s and m/s². Free to use, with no sign-up, no adverts and no personal tracking. A learning and checking tool: check the signs and your exam board’s value of g.

Start from an example

Fill in any three boxes and leave the other two empty. Pick a direction as positive and keep to it: a value in the opposite direction is negative.

Displacement s (m)Initial velocity u (m/s)Final velocity v (m/s)Acceleration a (m/s²)Time t (s)

Gravity g Significant figures

Result

300 mDisplacement s (found)0 m/sInitial velocity u (given)30.0 m/sFinal velocity v (given)1.50 m/s²Acceleration a (found)20.0 sTime t (given)

The gradient of the velocity graph is the acceleration, and the area under it is the displacement. The acceleration is constant, so the velocity graph is a straight line and the displacement graph is a parabola.

Working

Known values
Initial velocity u = 0 m/s
Final velocity v = 30 m/s
Time t = 20 s
Unknown: s and a.
Find the acceleration
Use v = u + at.
a = (v − u) / t = (30 − 0) / 20 = 1.50 m/s²
Find the displacement
Use s = ½(u + v)t.
s = ½(u + v)t = ½ × (0 + 30) × 20 = 300 m

Practise SUVAT questions

Press the button for a question with random numbers. Work it out on paper first, then type your answer.

How the equations of motion calculator works

The equations of motion describe a body moving in a straight line with constant acceleration. Each one leaves out one of the five quantities s, u, v, a and t:

v = u + at has no s.
s = ut + ½at² has no v.
v² = u² + 2as has no t.
s = ½(u + v)t has no a.
s = vt − ½at² has no u.

Enter any three values and the calculator picks the right equations, shows each substitution and then graphs the motion.

Sign convention

You choose the positive direction. For a ball thrown upward, take up as positive so that a = −9.81 m/s². For a dropped stone, take down as positive so that a = +9.81 m/s². A quantity in the opposite direction is negative.

Worked example

A cyclist speeds up from 4 m/s to 12 m/s while covering 100 m with constant acceleration. Find the acceleration and the time taken.

Known values
Displacement s = 100 m
Initial velocity u = 4 m/s
Final velocity v = 12 m/s
Unknown: a and t.
Find the time
Use s = ½(u + v)t.
t = 2s / (u + v) = 2 × 100 / (4 + 12) = 12.5 s
Find the acceleration
Use v = u + at.
a = (v − u) / t = (12 − 4) / 12.5 = 0.640 m/s²

The unknown quantities are a = 0.64 m/s² and t = 12.5 s.

Motion from rest at a constant 2 m/s²

A quick check on the equations: with u = 0 and a = 2 m/s², the velocity grows in step with the time and the distance grows with the square of the time.

Time	Velocity	Distance travelled
1 s	2 m/s	1 m
2 s	4 m/s	4 m
3 s	6 m/s	9 m
5 s	10 m/s	25 m
8 s	16 m/s	64 m
10 s	20 m/s	100 m

Which formulas does this use?

Constant-acceleration displacement, Mean acceleration, Braking distance, Total stopping distance with reaction time, Gravitational field and orbit.

Related calculators

SUVAT calculator, Free fall calculator, Projectile motion calculator, and the A-level physics equation calculator for the rest of the A-level equations.

Common questions

What are the equations of motion?

For constant acceleration they are v = u + at, s = ut + ½at², v² = u² + 2as, s = ½(u + v)t and s = vt − ½at². Each one leaves out one of the five quantities, which is how you choose between them.

Where do the equations of motion come from?

Acceleration is the gradient of the velocity–time graph, which gives v = u + at. The displacement is the area under that graph, a trapezium, which gives s = ½(u + v)t. Substituting one into the other gives the rest.

When do the equations of motion not work?

They only work when the acceleration is constant and the motion is along a straight line. If the acceleration changes, for example a car with changing engine force or a falling object with air resistance, you need calculus or a numerical method.

What if the object turns round?

The equations still work, as long as you keep the signs consistent. The displacement s is measured from the start, so it can drop back through zero. The calculator marks the instant the velocity changes sign on the graphs.

Is the calculator free, and does it keep my numbers?

It is free to use, with no sign-up, no adverts and no personal tracking. The numbers you type stay in your browser.

What do I assume about air resistance and g?

Air resistance is ignored, so these are the idealised answers used in GCSE, A-level and first-year university mechanics. Use g = 9.81 m/s² unless your exam board says 9.8 or 10. You can change g to compare other values or other worlds.