The Kinematic Equations Explained: How to Choose and Use SUVAT

The kinematic equations describe motion with constant acceleration. There are four, each linking four of the five quantities displacement (s), initial velocity (u), final velocity (v), acceleration (a) and time (t): v = u + at, s = ut + ½at², v² = u² + 2as and s = ½(u + v)t. To solve a problem, write down the three quantities you know, then choose the equation that leaves out the one you neither know nor need.

For example, a car braking from 30 m/s at 6 m/s² stops in v² = u² + 2as → 0 = 900 − 12s → s = 75 m.

The four equations

Some textbooks write v₀ or vᵢ for the initial velocity and Δx for displacement. The equations are the same.

Where they come from

The first equation is the definition of constant acceleration: velocity changes by a every second. The fourth says displacement is average velocity times time, and with constant acceleration the average velocity is exactly halfway between u and v. The other two follow by substituting one into the other to eliminate v or t.

How to solve a kinematics problem

1. Draw the situation and choose a positive direction.
2. List the five SUVAT variables and fill in the three you know, with signs.
3. Identify the unknown you need and the one you do not care about.
4. Choose the equation that skips the uncared-for variable.
5. Solve, then check the sign and size of the answer make sense.

Worked examples

Accelerating car

A car goes from rest to 60 mph (26.82 m/s) in 8.0 seconds. What is its acceleration, and how far does it travel?

Using s = ½(u + v)t avoids depending on the rounded acceleration. The acceleration calculator and velocity calculator solve these one step at a time.

Braking distance

A car traveling 30 m/s (about 67 mph) brakes at a steady 6.0 m/s². How far and how long does it take to stop?

Because distance depends on u², halving the speed to 15 m/s cuts the braking distance to a quarter, 18.75 m. Add the distance covered during the driver’s reaction time, about 1.5 s × 30 m/s = 45 m, and the total stopping distance is 120 m. The final velocity calculator uses v² = u² + 2as directly.

Free fall

A stone is dropped from a 45 m cliff. Ignoring air resistance, how long does it fall and how fast is it going at the bottom?

Standard gravity is defined as exactly 9.80665 m/s², a value adopted by the international General Conference on Weights and Measures; 9.81 m/s² is the usual working figure. The free fall calculator handles drops from any height.

Throwing a ball straight up

A ball leaves your hand moving upward at 15 m/s. How high does it go?

The way up and the way down are symmetric: the ball returns to your hand at 15 m/s, moving downward.

Sign conventions

Displacement, velocity and acceleration are vectors, so their signs carry direction. Pick one direction as positive and stick with it for the whole problem.

Most errors come from forgetting that gravity points down while the object moves up, or from treating “deceleration” as a positive number in the equations.

Units

Keep units consistent: meters, seconds and m/s² in SI, or feet, seconds and ft/s² in US units, where g ≈ 32.2 ft/s². Convert km/h to m/s by dividing by 3.6 (90 km/h = 25 m/s) and mph to m/s by multiplying by 0.44704.

When the equations do not apply

The four equations assume acceleration is constant in both size and direction. They do not apply to a car whose driver eases on and off the pedal, a skydiver approaching terminal velocity, or a planet in orbit. For those, you need calculus or numerical methods. For two-dimensional motion with constant gravity, such as a thrown ball, apply the equations separately to the horizontal and vertical directions; the projectile motion calculator does exactly that.

When you know three SUVAT values and want the other two in one step, the uniformly accelerated motion calculator solves the full set. Energy methods offer another route to the same answers; see how to calculate kinetic energy.