Final Velocity Calculator (v = u + at)
Solve the kinematics equation v = u + at for final velocity, initial velocity, acceleration, or time — with step-by-step working shown.
The first SUVAT equation
For motion with constant acceleration, the final velocity after time t is the initial velocity plus how much the acceleration has changed it:
- 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)
- t — time elapsed, in seconds (s) — always positive
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 for 5 seconds. What is its final velocity?
- Formula: v = u + at
- Substitute: v = 0 + 2 × 5
- Result: v = 10 m/s
Solving in reverse
The same formula rearranges to find initial velocity, acceleration, or time when the other three are known:
- Initial velocity: u = v − at — enter v, a, and t, leave u blank.
- Acceleration: a = (v − u) / t — enter v, u, and t, leave a blank.
- Time: t = (v − u) / a — enter v, u, and a, leave t blank.
For example, a car reaching 10 m/s from rest with acceleration 2 m/s² takes t = (10 − 0) / 2 = 5 s.
Common mix-ups
- Mixing up u and v. u is the velocity at the start of the interval you're measuring; v is the velocity at the end. Swapping them flips the sign of every solved acceleration or time.
- Forgetting the sign of deceleration. A car braking has acceleration opposite in sign to its direction of travel — treat it as negative if velocity is positive.
- Confusing velocity with displacement. v = u + at gives a speed, not a distance — a separate equation (s = ut + ½at²) is needed for how far the object travelled.
Where this shows up
v = u + at is the most-used kinematics equation: braking distance estimates, free-fall speed after a given time, and projectile motion all start here. It connects directly to force (F = ma) since acceleration is the shared quantity, and to momentum (p = mv) once the final velocity is known.
Frequently asked questions
- What is the formula for final velocity?
- The first SUVAT equation: v = u + at, where v is final velocity (m/s), u is initial velocity (m/s), a is acceleration (m/s²), and t is the elapsed time (s).
- How do I find acceleration from initial and final velocity?
- Rearrange the formula to a = (v − u) / t. Enter the final velocity, initial velocity, and time, leave acceleration blank, and the calculator solves it for you.
- Can initial velocity, final velocity, or acceleration be negative?
- Yes. A negative value just means the direction is opposite to whichever direction you called positive — for example, a decelerating car has acceleration opposite to its velocity. Only time must stay positive.
- Why does the calculator say "no valid time" for some inputs?
- Time is a physical duration and must be greater than 0. If the final velocity, initial velocity, and acceleration you entered would only be consistent with a negative or zero elapsed time, the calculator flags the inputs instead of showing an impossible answer.
- 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 seconds (s) for time. Convert other units (km/h, mph, minutes) to SI before entering them.
- How is this different from the displacement calculator?
- v = u + at finds velocity after a known time but says nothing about distance travelled. Use the displacement calculator (s = ut + ½at²) when you need how far the object moved.