Kinematics Solver — SUVAT Equations Explained and Solved
SUVAT equations describe motion under constant acceleration. They relate five variables — displacement, initial velocity, final velocity, acceleration, and time — through four equations. The free kinematics solver on PublicSoftTools finds any two unknowns from any three known values instantly.
The Four SUVAT Equations
| Equation | Variables used | Variable excluded |
|---|---|---|
| v = u + at | v, u, a, t | s |
| s = ut + ½at² | s, u, a, t | v |
| v² = u² + 2as | v, u, a, s | t |
| s = ½(u + v)t | s, u, v, t | a |
Each equation uses four of the five variables and excludes one. This means you can always solve for two unknowns as long as you know three of the five values.
How to Use the Solver
- Open the kinematics solver.
- Enter the three values you know. Leave the other two fields blank.
- The solver automatically applies the correct equations and fills in the two unknown values.
- Results appear in color-highlighted fields to distinguish solved values from entered values.
Worked Examples
Free fall from rest
A ball is dropped from a 45 m building. Find the time to hit the ground and impact velocity. Known: s = 45 m, u = 0 m/s, a = 9.81 m/s². Enter these three values — the solver gives t ≈ 3.03 s and v ≈ 29.7 m/s.
Braking distance
A car traveling at 30 m/s (108 km/h) brakes to a stop with a deceleration of 8 m/s². Known: u = 30 m/s, v = 0 m/s, a = −8 m/s². The solver gives: s ≈ 56.25 m (stopping distance) and t ≈ 3.75 s. This matches the real-world figure that highway braking from 108 km/h requires about 56 meters.
Constant acceleration
A train accelerates from 0 to 25 m/s over 500 m. Known: u = 0 m/s, v = 25 m/s, s = 500 m. The solver gives: a = 0.625 m/s² and t = 40 s.
Sign Conventions
The sign convention is crucial. Choose a positive direction — usually the initial direction of motion. Then:
- Displacement in the positive direction is positive; backward displacement is negative.
- Velocity in the positive direction is positive; velocity in the opposite direction is negative.
- Acceleration in the positive direction is positive; deceleration (opposing motion) is negative.
For a ball thrown upward, choose up as positive. Then: u = +20 m/s, a = −9.81 m/s² (gravity opposes the upward motion).
Common Questions
What if the solver doesn't find a solution?
This can happen if the three values are physically inconsistent — for example, if final velocity is faster than initial velocity but acceleration is negative. Check your sign conventions and ensure the scenario is physically realistic.
Can SUVAT equations be used for variable acceleration?
No. SUVAT applies only to constant acceleration. For variable acceleration (like a rocket with changing thrust), you need calculus-based kinematics or numerical integration.
Solve Kinematics Now
Enter any 3 of 5 SUVAT variables — displacement, velocity, acceleration, or time — to solve for the rest.
Open Kinematics Solver