Kinematics (SUVAT) Calculator

A car goes from 0 to 100 km/h in 8 s. Convert first: 100 km/h = 100 ÷ 3.6 = 27.78 m/s. With u, v and t known, a = (v − u) ÷ t = 27.78 ÷ 8 = 3.47 m/s² (0.354 g), and s = ½(u + v)t = ½ × 27.78 × 8 = 111.1 m.

A ball thrown straight up at 15 m/s. Take up as positive, so a = −9.80665 m/s². At the top v = 0, so t = (0 − 15) ÷ −9.80665 = 1.53 s and s = (0 − 15²) ÷ (2 × −9.80665) = 11.47 m. Asking when it is 10 m up gives a quadratic with two positive roots, 0.98 s and 2.08 s: once rising, once falling.

The common mistake: a positive acceleration for braking. A car braking from 30 m/s at 6 m/s², with a entered as +6, gives s = (0 − 30²) ÷ (2 × 6) = −75 m: a negative stopping distance for a car that went forward. With the sign right, a = −6 m/s², s = (0 − 900) ÷ (−12) = 75 m and t = 5 s.

The equation s = ½(u + v)t is older than Newton. In the 1330s a group of scholars at Merton College, Oxford (William Heytesbury, Richard Swineshead and John Dumbleton) stated the mean speed theorem: a body accelerating uniformly covers the same distance as one moving steadily at its average speed. Around 1350 Nicole Oresme in Paris proved it with a diagram in which the distance is the area of a triangle under a speed line, the same picture as the velocity–time graph above.

Galileo Galilei tested the idea with balls rolling down inclined planes and published the result in 1638 in Two New Sciences: distance from rest grows with the square of time, so the distances in successive equal intervals go 1 : 3 : 5 : 7. Newton’s second law (1687) then explained why: a constant force gives a constant acceleration. The letters s, u, v, a, t are the British school convention, which is where the name SUVAT comes from.

All five equations assume the acceleration is constant. Air resistance, a car changing gear or a rocket burning fuel break that assumption; for those the motion has to be split into short pieces or integrated numerically.

This calculator solves straight-line motion with constant acceleration for all ten pairs of unknowns. It chooses the SUVAT equation that leaves out the quantity you didn’t give, solves the quadratic for time when it has to and explains which roots are physical, and checks the answer against v² = u² + 2as. Every field takes its own unit, so km/h, feet and g can be mixed; the working shows each conversion to SI.

The velocity–time graph shades the displacement, above the axis for forward motion and below it for backward motion, and the calculator reports the distance travelled separately when the object turns round. Everything runs in your browser; nothing is sent anywhere.

Related tools: Projectile Motion Calculator, Kinetic & Potential Energy Calculator, and Speed, Distance & Time Calculator.