Last updated: 2026-06-14

Kinematic Equations

The kinematic equations describe motion under constant acceleration. They connect five quantities — initial velocity ($u$), final velocity ($v$), acceleration ($a$), time ($t$), and displacement ($s$) — through four equations, each of which omits one variable. Knowing any three of the five lets you find the other two. In UK physics these are known as the SUVAT equations, after the symbols involved; elsewhere they are simply called the equations of motion.

This calculator selects the right equation for the quantities you have and solves for the final velocity, the displacement, or the acceleration.

The four equations

The choice of equation comes down to which variable you do not have. If a problem gives no time, the third equation ($v^2 = u^2 + 2 a s$) is the one to use; if it gives no acceleration, the fourth works directly from the average of the two velocities.

Worked example

A car accelerates uniformly from rest to 27.8 m/s (100 km/h) in 8 seconds. The acceleration is:

and the distance covered while reaching that speed is found from the average velocity:

Selecting "Acceleration from u, v, t" with $u = 0$, $v = 27.8$ m/s, and $t = 8$ s gives the acceleration, and "Displacement from u, v, t" gives the distance.

Sign conventions

The equations are vector relationships in one dimension, so signs matter. Choose a positive direction and keep it consistent: a velocity opposing that direction is negative, and a deceleration is a negative acceleration. For an object thrown straight up, taking upward as positive makes the acceleration of gravity $-9.81$ m/s². Getting the signs right is the most common source of error in kinematics problems.

When constant acceleration applies

These equations hold only while acceleration stays constant. They describe free fall near Earth's surface, a vehicle braking steadily, or a block sliding down a frictionless incline. They do not apply when the force — and therefore the acceleration — changes during the motion, as with air resistance or a varying engine thrust; those situations need calculus. For two-dimensional motion such as projectiles, the equations are applied separately to the horizontal and vertical components, since gravity affects only the vertical direction.