Final Velocity Calculator
Starting from rest and accelerating at 3 m/s² for 8 seconds. To find velocity from distance instead of time, switch the velocity calculator to another solve-for pairing.
Final velocity v
- Final velocity v From v = u + at: v = 0 + 3 × 8 = 24 m/s.
- 24 m/s
- Displacement s From s = ut + ½at²: s = 0 × 8 + ½ × 3 × 8² = 96 m.
- 96 m
- Knowns used (SI) Everything is converted to meters, seconds and meters per second before the equations run, so mixed units cannot leak into the arithmetic.
- u = 0 m/s, a = 3 m/s², t = 8 s
- Average velocity ½(u + v). Under constant acceleration, and only then, the average velocity is the mean of the start and end velocities, which is why s = ½(u + v)t works.
- 12 m/s
- Acceleration in g Divided by standard gravity, 9.80665 m/s². A useful sense check: a hard road-car launch is a fraction of 1 g, and free fall is exactly 1 g.
- 0.3059 g
- Final velocity in other units
- 24 m/s 86.4 km/h 53.69 mph 78.74 ft/s
- Displacement in other units
- 96 m 0.096 km 315 ft 0.05965 mi
| Equation | Leaves out | Left side | Right side | Used here |
|---|---|---|---|---|
| v = u + at | s | 24 | 24 | yes |
| s = ut + ½at² | v | 96 | 96 | yes |
| v² = u² + 2as | t | 576 | 576 | |
| s = ½(u + v)t | a | 96 | 96 | |
| s = vt − ½at² | u | 96 | 96 |
- The v and s boxes are ignored: those are the two being solved for.
- These equations assume the acceleration is constant for the whole interval. If it changes (a car shifting gear, anything with real air resistance), split the motion into pieces that are each roughly constant.
- Results are shown to four significant figures. Round the answer to the precision of your least precise measurement.
How the Kinematics Calculator works
The formula for velocity under constant acceleration is v = u + at: final velocity equals initial velocity plus acceleration times time. It is the definition of acceleration rearranged. If acceleration is the change in velocity per second, then after t seconds the velocity has changed by a·t.
Worked example, the one above: starting from rest (u = 0) and accelerating at 3 m/s² for 8 s, the final velocity is v = 0 + 3 × 8 = 24 m/s. That is 86.4 km/h or 53.7 mph. The distance covered on the way is s = ut + ½at² = ½ × 3 × 8² = 96 m, and the average velocity is ½(0 + 24) = 12 m/s, which times 8 s is the same 96 m.
When time is not given, use v² = u² + 2as instead. A car braking from speed u to rest over a distance s has v = 0, so u² = −2as, and the deceleration follows. To use that here, choose "Time t and acceleration a (know s, u, v)", or whichever pairing names the knowns you have.
Velocity and speed are not the same. Velocity has a direction, so it can be negative: a ball thrown upward at 15 m/s with up as positive has v = 15 − 9.80665 × 2 = −4.613 m/s after two seconds, meaning it is coming back down at 4.613 m/s. Its speed is the size of that, 4.613 m/s. The formula handles the reversal automatically as long as the signs are consistent; a negative answer is information, not an error.
Average velocity is a different formula. For any motion, average velocity is displacement divided by time. Only under constant acceleration does it also equal ½(u + v), the mean of the start and end velocities. A trip that crawls through traffic and then speeds on the open road has an average velocity of total distance over total time, and ½(u + v) would be wrong.
Units. The formula works in any consistent set: m/s with m/s² and s, or ft/s with ft/s² and s. The trap is mixing them: a speed in km/h with an acceleration in m/s² and a time in seconds gives nonsense unless one is converted first. The calculator converts everything to SI before solving and shows the answer in your unit, in m/s, and in the other common speed units.
Kinematics Calculator FAQ
What is the formula for final velocity?
v = u + at, where u is the initial velocity, a the constant acceleration and t the time. From rest at 3 m/s² for 8 s, v = 24 m/s. If you know distance rather than time, use v² = u² + 2as and take the square root.
Can velocity be negative?
Yes. Velocity includes direction, so a negative value means moving in the direction you chose as negative. A ball thrown upward has a negative velocity on the way down if up is positive. Speed is the size of the velocity and is never negative.
How do I find velocity without time?
Use v² = u² + 2as, the one kinematic equation without t. A car accelerating from rest at 3 m/s² over 96 m reaches v = √(2 × 3 × 96) = 24 m/s. If v² comes out negative, the object turns back before covering that distance.