Kinematics Calculator (SUVAT)
Enter any three SUVAT variables to solve for the remaining two. Supports displacement, velocity, acceleration, and time.
What are kinematics equations?
Kinematics is the branch of physics that describes motion without considering the forces that cause it. The four kinematic equations relate five variables: displacement (s), initial velocity (u), final velocity (v), acceleration (a), and time (t). Given any three of these five values, you can calculate the remaining two.
These equations assume constant (uniform) acceleration, which covers many real-world scenarios: objects in free fall (constant gravitational acceleration), vehicles accelerating or braking at a steady rate, and objects sliding on surfaces with constant friction. For non-constant acceleration, calculus-based methods are needed.
The four equations
- v = u + at — Final velocity from initial velocity, acceleration, and time.
- s = ut + (1/2)at squared — Displacement from initial velocity, time, and acceleration.
- v squared = u squared + 2as — Final velocity from initial velocity, acceleration, and displacement (no time needed).
- s = (u + v)/2 * t — Displacement from average velocity and time.
How to use this tool
Enter any three of the five kinematic variables and the calculator solves for the remaining two. Select which variable to solve for, or let the tool auto-detect based on which fields are filled. Results include both the numerical answer and the equation used.
Free fall as a special case
Free fall is kinematics with a = g (approximately 9.81 m/s squared downward). A dropped object (u = 0) falls a distance s = (1/2)gt squared. After 1 second: 4.9m. After 2 seconds: 19.6m. After 3 seconds: 44.1m. The velocity increases linearly: 9.8 m/s after 1 second, 19.6 m/s after 2 seconds, and so on (ignoring air resistance).
Worked example
A car accelerates from rest at 3 m/s² for 5 seconds. Using v = u + at, final velocity is 0 + 3 × 5 = 15 m/s. The distance is s = ut + ½at², which is 0 + ½ × 3 × 5² = 37.5 m. These SUVAT equations assume constant acceleration, so they describe a car pulling away smoothly but not one whose acceleration changes moment to moment.
Common mistakes
The most common error is applying the constant-acceleration equations to motion where acceleration is not constant — they simply do not hold then. Another is sign confusion: choose a positive direction and keep it, so a ball thrown upward has positive velocity but negative (downward) acceleration of −9.8 m/s². Mixing units, such as combining km/h with seconds, is a third frequent slip; convert to metres and seconds first.
Frequently asked questions
Do the equations work for objects moving upward?
Yes. Use a positive initial velocity for upward motion and negative acceleration (gravity pulling down). The equations automatically handle the object slowing down, stopping at the peak, and falling back. The peak height occurs when v = 0, and the total flight time is twice the time to reach the peak (for symmetric trajectories).
What happens when acceleration is not constant?
These equations only apply when acceleration is constant. For variable acceleration (like a rocket burning fuel, or an object encountering increasing air resistance), you need to use calculus — specifically, integrate the acceleration function over time. Numerical methods can also approximate the motion in small time steps.
What is the difference between speed and velocity?
Speed is a scalar — just how fast something moves. Velocity is a vector that also includes direction. A car going around a roundabout at constant speed has continuously changing velocity because its direction changes, which means it is accelerating even though the speedometer reads steady.