The SUVAT equations (kinematic equations)
In uniformly accelerated motion in a straight line the acceleration is constant. Five quantities describe it: initial velocity (v₀ or u), final velocity (v), acceleration (a), time (t) and displacement (x or s), which is why they are known as the SUVAT equations. Each equation links four of them, so any three values let you calculate the other two:
| Equation | Missing quantity |
|---|---|
| v = v₀ + a·t | x |
| x = v₀·t + ½·a·t² | v |
| v² = v₀² + 2·a·x | t |
| x = (v₀ + v)/2 · t | a |
| x = v·t − ½·a·t² | v₀ |
The trick for choosing an equation: look at which quantity you are neither given nor asked for, and use the equation that does not contain it.
Example · braking distance
A car at 25 m/s (90 km/h, about 56 mph) brakes to a stop in 50 m. Time is neither given nor asked for, so use v² = v₀² + 2ax: 0 = 625 + 2·a·50 → a = −6.25 m/s². Then t = (v − v₀)/a = 25/6.25 = 4 s.
Signs and free fall
Choose a positive direction and stick to it: quantities pointing the other way get a minus sign. In free fall and vertical throws, if you take "up" as positive, a = −g = −9.81 m/s². If the object is slowing down, the acceleration has the opposite sign to the velocity.
With a = 0 the motion has constant velocity: v does not change and x = v·t.
Frequently asked questions
Why do I sometimes get two times?
Because x = v₀t + ½at² is quadratic in t. In a vertical throw, for example, the object passes the same height on the way up and on the way down. The calculator shows the first and tells you about the second.
Is x the distance travelled?
It is the displacement: final position minus initial position. It equals the distance travelled only if the object does not change direction.
How do I convert km/h or mph to m/s?
Divide km/h by 3.6, or multiply mph by 0.44704. For other units use the unit converter.