Kinematics Calculator

Motion with constant acceleration is described by five quantities: displacement s, initial velocity u, final velocity v, acceleration a and time t. Know any three and the other two follow. This calculator does exactly that: fill in any three boxes, leave two blank, and it picks the right equations of motion and solves for the missing pair, in any mix of units. It lists which equations it used, so you can follow the working, and plots position and velocity over the motion.

It handles speeding up, slowing down and objects thrown upward, as long as you keep a consistent sign convention: choose one direction as positive and give velocities and acceleration pointing the other way a minus sign.

v = u + at

s = ut + 1 over 212at²

s = vt − 1 over 212at²

v² = u² + 2as

s = (u + v)t ÷ 2

Each equation leaves out one of the five quantities, which is why any three known values are enough. The calculator chooses the pair of equations that avoids the two unknowns. These equations assume the acceleration stays constant throughout; for changing acceleration, break the motion into stages and solve each one.

Braking distance. A car at 20 m/s (72 km/h) braking at 5 m/s² stops in 4 s over 40 m.

A dropped stone. After 3 s of free fall from rest it has fallen 44.13 m and is moving at 29.42 m/s.

Take-off. An aircraft that must reach 80 m/s within 1,200 m of runway needs an average acceleration of 2.667 m/s², taking 30 s.

A car from 0 to 100 km/h in 8 s accelerates at 3.472 m/s² and covers 111.1 m.

Distance to stop at a constant 7 m/s² of braking, ignoring reaction time:

Doubling speed quadruples the braking distance, from v² = u² + 2as. Add the distance traveled during the driver's reaction time, typically 1 to 2 seconds at full speed, for the total stopping distance.

Most mistakes in these problems come from signs. Pick a positive direction, often the direction of the initial motion or upward for vertical problems, and stick to it. A ball thrown straight up at 15 m/s with gravity pulling down has u = +15 m/s and a = −9.81 m/s²; at the top v = 0, and when it returns to the hand s = 0 and v = −15 m/s. If a result comes out negative, it points opposite to your positive direction, which is often exactly right.

Many real problems have stages with different accelerations: a car speeds up, cruises, then brakes; a rocket burns its engine, then coasts upward under gravity alone. Solve each stage separately with this calculator and pass the final velocity of one stage on as the initial velocity of the next. Distances and times add across the stages.

For example, a train that accelerates from rest at 0.5 m/s² for 60 s reaches 30 m/s after covering 900 m. If it then cruises for two minutes it covers another 3,600 m, and if it finally brakes at 1 m/s² it needs 30 s and 450 m to stop: 4,950 m in 210 s in total. Writing down u, v, a, s and t for each stage in a small table keeps the bookkeeping clear and makes errors easy to spot.

For problems that only involve velocity change and time, the acceleration calculator is quicker. For objects launched at an angle, which combine steady horizontal motion with accelerated vertical motion, use the projectile motion calculator, and for straight drops the gravity calculator.