# What is the final velocity?

Solves the constant-acceleration equations of motion: the final velocity, initial velocity, acceleration, time or displacement from any three of them, with the average velocity.

- Page: https://www.acalculator.org/physics/velocity-calculator
- JSON spec: https://www.acalculator.org/physics/velocity-calculator.json
- Version: 0be65ab79d79

## Default answer

Example with the default inputs (Initial velocity (v₀) 70 m/s, Acceleration (m/s²) -1.5, Time 40 s): Starting at 70 m/s with an acceleration of -1.5 m/s² for 40 s, the final velocity is 10 m/s.

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| u | Initial velocity (v₀) | The velocity at the start; negative means moving the other way. |
| v | Final velocity (v) | The velocity at the end; negative means moving the other way. |
| a | Acceleration (m/s²) | The constant acceleration in metres per second squared; negative slows a forward motion. |
| t | Time | How long the motion lasts. |
| s | Displacement | The change in position over the time; negative means it ended behind the start. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| u | Initial velocity | The velocity at the start. |
| v | Final velocity | The velocity at the end of the time: v = v₀ + a × t. |
| a | Acceleration (m/s²) | The constant acceleration in metres per second squared; negative slows a forward motion. |
| t | Time | How long the motion lasts. |
| s | Displacement | The change in position over the time; negative means it ended behind the start. |
| average | Average velocity | Displacement ÷ time, which under constant acceleration is (v₀ + v) ÷ 2. |

## Method

v = v₀ + a t; s = ½ (v₀ + v) t; s = v₀ t + ½ a t²; s = v t − ½ a t²; v² = v₀² + 2 a s. Average velocity = s ÷ t = (v₀ + v) ÷ 2.

## Assumptions

- The acceleration is constant over the whole time, and the motion is along one straight line; a negative value points the other way.
- When two answers fit (a velocity from v² = v₀² + 2as, or a time from a quadratic), the page shows both.

## Worked examples

1. u = 70, a = -1.5, t = 40 gives v = 10, s = 1,600, average = 40. Source: OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Example 3.7: v = 70.0 m/s + (−1.50 m/s²)(40.0 s) = 10.0 m/s).
2. u = 0, a = 26, t = 5.56 gives s = 401.8768, v = 144.56. Source: OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Example 3.8: x = ½ (26.0 m/s²)(5.56 s)² = 402 m).
3. u = 0, a = 26, s = 402 gives v = 144.582157, t = 5.560852. Source: OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Example 3.9: v = √(2 × 26.0 m/s² × 402 m) = 145 m/s).
4. u = 20, v = 5, s = 50 gives a = -3.75, t = 4, average = 12.5. Source: OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration (Equations 3.10 to 3.14), https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (Equation 3.14).

## FAQ

### How do I calculate final velocity?

Add the acceleration times the time to the initial velocity: v = v₀ + a t. A jet landing at 70 m/s that slows at 1.5 m/s² for 40 s ends at 70 − 1.5 × 40 = 10 m/s.

### How do I find velocity without time?

Use v² = v₀² + 2as. A dragster from rest at 26 m/s² over 402 m reaches √(2 × 26 × 402) ≈ 144.6 m/s. The square root has two signs; the page shows both when both fit.

### What is the difference between speed and velocity?

Velocity has a direction, so it can be negative here: −5 m/s means 5 m/s the other way. Speed is the size of the velocity. For average speed over a trip, see the speed calculator.

### How do I calculate average velocity?

Divide the displacement by the time. With constant acceleration it is also the mean of the start and end velocities, (v₀ + v) ÷ 2. The jet above averages (70 + 10) ÷ 2 = 40 m/s and covers 1,600 m.

### Why does the page show two answers?

Some inputs fit two motions. A ball thrown up at 10 m/s with a = −9.8 m/s² is 4 m up twice: once rising and once falling. The page shows both times, or both signs of the velocity.

### What units can I use?

Velocities in m/s, km/h, mph, ft/s or knots; displacement in m, km, ft or miles; time in s, min or h. Acceleration is in m/s²; for ft/s², multiply by 0.3048 first.

## Sources

- OpenStax, University Physics Volume 1, §3.4 Motion with Constant Acceleration: Equations 3.10 to 3.14, Examples 3.7 (landing jet), 3.8 and 3.9 (dragster). https://openstax.org/books/university-physics-volume-1/pages/3-4-motion-with-constant-acceleration (retrieved 2026-10-01)
- NIST Special Publication 811, Appendix B: 1 ft = 0.3048 m, 1 mi = 1,609.344 m, 1 nautical mile = 1,852 m exactly. https://www.nist.gov/pml/special-publication-811 (retrieved 2026-10-01)