Velocity-Squared Calculator (v² = u² + 2as)
Solve the SUVAT equation v² = u² + 2as for final velocity, initial velocity, acceleration, or displacement — no time needed.
The time-free SUVAT equation
For motion with constant acceleration, this equation connects final velocity, initial velocity, acceleration, and displacement — with no time term at all:
- v — final velocity, in metres per second (m/s)
- u — initial velocity, in metres per second (m/s)
- a — acceleration, in metres per second squared (m/s²)
- s — displacement, in metres (m)
Enter any three values and leave the fourth blank; the calculator solves for it and shows the substitution step.
Worked example
A car starts from rest (u = 0 m/s) and accelerates at 2 m/s² over a displacement of 25 m. What is its final velocity?
- Formula: v² = u² + 2as
- Substitute: v = √(0² + 2 × 2 × 25)
- Result: v = 10 m/s
Solving in reverse
The same formula rearranges to find initial velocity, acceleration, or displacement:
- Initial velocity: u = √(v² − 2as) — enter v, a, and s, leave u blank.
- Acceleration: a = (v² − u²) / 2s — enter v, u, and s, leave a blank.
- Displacement: s = (v² − u²) / 2a — enter v, u, and a, leave s blank.
For example, a car reaching 10 m/s from rest with acceleration 2 m/s² needs s = (10² − 0²) / (2 × 2) = 25 m.
Common mix-ups
- Forgetting to square v and u. The equation uses v² and u², not v and u directly — a common slip when rearranging by hand.
- Sign errors under the square root. A negative acceleration combined with a positive displacement (or vice versa) can make u² + 2as negative — check the signs before assuming a mistake.
- Reaching for this equation when time is known. If you already have t, v = u + at or s = ut + ½at² is usually simpler.
Where this shows up
v² = u² + 2as is the standard way to find braking distance from speed (or vice versa) without needing to know how long braking takes, and to find impact speed after falling a known height. Once the final velocity is known, it connects to kinetic energy (KE = ½mv²) and momentum (p = mv).
Frequently asked questions
- What is the formula for v² = u² + 2as?
- The third SUVAT equation relates final velocity v (m/s), initial velocity u (m/s), acceleration a (m/s²), and displacement s (m) — without needing the time t at all.
- When should I use this equation instead of v = u + at?
- Use v² = u² + 2as whenever time isn't given or isn't needed — for example, finding a car's speed after a known braking distance, or the displacement needed to reach a target speed.
- Why does the calculator sometimes say "no real solution"?
- Solving for v or u takes a square root of u² + 2as (or v² − 2as). If that quantity is negative — for example, a positive initial speed with a large enough opposing (negative) acceleration over that displacement — there's no real velocity that satisfies the equation, so the calculator reports it instead of guessing.
- Can acceleration or displacement be negative?
- Yes. A negative acceleration means slowing down or reversing, and a negative displacement means net movement opposite to your chosen positive direction — both are valid inputs.
- What units does this calculator use?
- SI units throughout — metres per second (m/s) for velocity, metres per second squared (m/s²) for acceleration, and metres (m) for displacement.
- How is this related to the displacement calculator?
- The displacement calculator (s = ut + ½at²) needs time as an input; this one eliminates time entirely by combining it algebraically with the other SUVAT equations, which is useful when time isn't part of the problem.