Velocity calculator: from distance and time, or with acceleration
Velocity is displacement divided by time: v = s ÷ t, in metres per second, so 100 m in 20 s is 5 m/s. If the object is speeding up or slowing down at a steady rate, use v = u + at for its final velocity, or v² = u² + 2as if you don't know the time. This velocity calculator is our SUVAT Calculator: type any three of displacement, starting velocity, final velocity, acceleration and time, and it finds the other two.
This small version starts on a car doing 8 m/s that accelerates at 2 m/s² for 5 s: its final velocity is 18 m/s. For graphs and full working, open the full SUVAT calculator.
- Displacements = 65 m
- Final velocityv = 18 m/s
Uses s = u t + \frac{1}{2} a t^2
How to find velocity with the calculator
The letters are s (displacement), u (initial velocity), v (final velocity), a (acceleration) and t (time). Fill in the three you know, leave the other two blank, and the calculator says which equation it used and why. Which three you type depends on the question:
| You know | Type | It uses |
|---|---|---|
| Distance and time, at a steady speed | s, t and a = 0 | v = s ÷ t |
| Starting velocity, acceleration and time | u, a and t | v = u + at |
| Starting velocity, acceleration and distance | u, a and s | v² = u² + 2as |
| Distance, time and starting velocity | s, t and u | s = ½(u + v)t |
Each box has its own unit, so you can type km/h, mph or km and the calculator converts them. The full calculator also draws a velocity-time graph with the displacement shaded.
Velocity formulas
Velocity from distance and time
Strictly, s is the displacement: how far you end up from the start, in a direction. Over a whole journey, total displacement ÷ total time is the average velocity.
Final velocity with constant acceleration
The first says the final velocity is the starting velocity plus the velocity gained, a for every second of t. The second leaves out time, so use it when the question gives a distance instead. Both only work while the acceleration is constant. The SUVAT equations guide covers all five equations and how to choose between them.
Worked examples you can check in the calculator
Example 1: constant velocity
A cyclist covers 100 m in 20 s at a steady pace. v = 100 ÷ 20 = 5 m/s. In the calculator, type s = 100, t = 20 and a = 0, and it gives u = v = 5 m/s, since with no acceleration the velocity never changes.
Example 2: final velocity from v = u + at
A car doing 8 m/s accelerates at 2 m/s² for 5 s, the small calculator's starting values. v = 8 + 2 × 5 = 18 m/s. The calculator also gives the distance covered, 65 m.
Example 3: final velocity without the time
A car at 10 m/s accelerates at 1.5 m/s² over 100 m. There's no time, so use v² = u² + 2as:
So v = √400 = 20 m/s. Type u = 10, a = 1.5 and s = 100 to check, and the calculator adds the time, 6.67 s.
Example 4: velocity of a falling object
A stone is dropped from 20 m. Ignoring air resistance it starts at rest (u = 0) and accelerates at about 9.8 m/s². With u = 0, a = 9.8 and s = 20:
It hits the ground at 19.8 m/s, after 2.02 s.
Example 5: a negative velocity
A ball is thrown straight up at 15 m/s. Taking up as positive, a = −9.8 m/s². After 2 s, v = 15 + (−9.8) × 2 = −4.6 m/s. The minus sign means it is now moving downwards at 4.6 m/s, even though it is still 10.4 m above where it was thrown. That direction is what makes velocity different from speed.
Velocity vs speed
Speed is how fast. Velocity is how fast and in which direction, so it can be negative. Over a journey, average speed uses the total distance travelled and average velocity uses the displacement. Walk 400 m east and then 300 m west, taking 500 s in all: you've travelled 700 m but ended up only 100 m east of the start.
| Average speed | Average velocity | |
|---|---|---|
| Uses | Total distance, 700 m | Displacement, 100 m east |
| Working | 700 ÷ 500 | 100 ÷ 500 |
| Answer | 1.4 m/s | 0.2 m/s east |
Velocity on a velocity-time graph
On a velocity-time graph you read the velocity straight off the vertical axis. The gradient (slope) is the acceleration and the area underneath is the displacement. Try reading one, then press Mark to check your answer in the Laws of Motion course.
Read an acceleration (slope) or a distance (area) off a random velocity-time graph.
Press Mark to have your answer marked in the Laws of Motion course.
Common mistakes
- Mixed units. Change km/h to m/s by dividing by 3.6: 90 km/h is 25 m/s. Or choose km/h as the unit in the calculator.
- Using v = s ÷ t when the speed is changing. That gives the average velocity, not the final one. For the final velocity with steady acceleration, use v = u + at.
- Missing a hidden zero. "From rest" and "dropped" mean u = 0. "Comes to a stop" means v = 0.
- Dropping the sign. Choose a positive direction and stick to it. Motion the other way is negative.
- Forgetting the square root in v² = u² + 2as. The 400 in example 3 is v², not v.
Common questions
What is the formula for velocity?
How do you find final velocity without time?
What is the unit of velocity?
Can velocity be negative?
How do you find average velocity?
Revision cards
Read the front, say your answer, then flip to check.
Where to go next
- Solve it
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