A car at 72 km/h
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m = 1500 kg, v = 72 km/hConvert the speed first: 72 km/h = 72 · 1000/3600 = 20 m/s. Then p = 1500 · 20 = 30,000 kg·m/s. Putting 72 straight in would give 108,000, 3.6 times too much.
Momentum Calculator
Momentum, p = mv, and the impulse of a force, J = FΔt.
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Want to solve these without the calculator?
The Coddy physics course teaches the method: you solve each problem in steps on an interactive board, and it tells you which step went wrong.
The momentum formula, and what it says
Momentum is how much motion an object carries: its mass times its velocity, p = mv. With m in kilograms and v in metres per second it comes out in kg·m/s. A 0.16 kg cricket ball bowled at 30 m/s has 0.16 · 30 = 4.8 kg·m/s; a 1500 kg car at 20 m/s has 30,000 kg·m/s.
Momentum has a direction, the direction of the velocity. Pick one way as positive and anything moving the other way gets a negative velocity and a negative momentum. To change an object's momentum a force has to act on it for some time, and that product is the impulse, J = FΔt. The impulse-momentum theorem says the impulse equals the change in momentum: FΔt = m(v − v₀).
The calculator has two tabs. The momentum tab solves p = mv for whichever of p, m and v you leave blank. The impulse tab links the force and contact time to the mass and the two velocities: give a mass, a starting and a final velocity and a time, and it finds the impulse and then the average force; give a force, a time, a mass and a starting velocity, and it finds the impulse and the final velocity. Velocities in km/h or mph and times in ms are converted first, and minus signs are kept.
What to notice
- p = mv gives kg·m/s only when the mass is in kilograms and the velocity in m/s. A newton second, N·s, is the same unit.
- Momentum has a direction. Moving left at 5 m/s and moving right at 5 m/s are opposite momenta, so one of them gets a minus sign.
- Impulse is force times time, J = FΔt, and it equals the change in momentum, m(v − v₀). The same change can come from a big force for a short time or a small force for a long time.
- A negative impulse or force means it acts against the direction you called positive, as when something is brought to a stop.
- Momentum grows in step with the speed; kinetic energy, ½mv², grows with its square. Twice as fast means twice the momentum but four times the energy.
How to calculate momentum and impulse
Put the mass in kilograms
Grams divide by 1000, pounds multiply by 0.4536. A 160 g ball is 0.16 kg.
Put the velocity in metres per second, with a sign
km/h divide by 3.6, mph multiply by 0.44704. Choose a positive direction and give anything moving the other way a minus sign.
Multiply for the momentum
p = m · v. A 1500 kg car at 72 km/h, which is 20 m/s, has p = 1500 · 20 = 30,000 kg·m/s.
For an impulse, take the change in velocity
J = m · (v − v₀), final minus initial. Then the average force is F = J / Δt, with the contact time in seconds.
Momentum of everyday things
Rounded values from p = mv, all moving in the positive direction. A light bullet carries less momentum than a person walking.
| Object | Mass | Velocity | Momentum |
|---|---|---|---|
| Pistol bullet | 8 g | 360 m/s | 2.88 kg·m/s |
| Cricket ball | 0.16 kg | 30 m/s | 4.8 kg·m/s |
| Baseball pitch | 0.145 kg | 40 m/s | 5.8 kg·m/s |
| Person walking | 70 kg | 1.4 m/s | 98 kg·m/s |
| Sprinter | 80 kg | 10 m/s | 800 kg·m/s |
| Car in town | 1,500 kg | 50 km/h | 20,800 kg·m/s |
| Car on a highway | 1,500 kg | 100 km/h | 41,700 kg·m/s |
| Loaded truck | 40,000 kg | 90 km/h | 1,000,000 kg·m/s |
Worked examples
Kicking a football
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m = 0.45 kg, v₀ = 0 m/s, v = 25 m/s, Δt = 0.01 sThe impulse is the change in momentum: J = 0.45 · (25 − 0) = 11.25 N·s. The foot delivers it in 0.01 s, so the average force is F = J / Δt = 11.25 / 0.01 = 1125 N. The calculator finds both from these four values.
Stopping a runner
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m = 70 kg, v₀ = 10 m/s, v = 0 m/s, Δt = 0.5 sJ = 70 · (0 − 10) = −700 N·s, and F = −700 / 0.5 = −1400 N. The minus sign says the force points against the motion. Stop in 0.05 s instead and the force is −14,000 N: the same impulse over a tenth of the time takes ten times the force, which is why padding and airbags work.
A ball that bounces back
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m = 0.5 kg, v₀ = 10 m/s, v = −8 m/s, Δt = 0.02 sThe ball comes back the other way, so its final velocity is −8 m/s. J = 0.5 · (−8 − 10) = −9 N·s and F = −9 / 0.02 = −450 N. Dropping the sign and using 8 − 10 would give only −1 N·s.
The velocity from a momentum
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p = 30,000 kg·m/s, m = 1500 kgRearrange for v: v = p / m = 30,000 / 1500 = 20 m/s. Set the velocity's unit to km/h before solving and the answer comes out as 72 km/h.
Common mistakes
- Using km/h or mph in the formula. Convert to m/s first, or the answer is out by a factor of 3.6 (km/h) or about 2.2 (mph).
- Using grams instead of kilograms. 160 g is 0.16 kg; using 160 makes the momentum a thousand times too big.
- Dropping the direction. A ball that bounces back has a negative final velocity; treating it as positive makes the change in momentum far too small.
- Subtracting the wrong way. The change in velocity is final minus initial, v − v₀. Swapping them flips the sign of the impulse and the force.
- Using a contact time in milliseconds as if it were seconds. 10 ms is 0.01 s; choose ms as the unit and the calculator converts it.
- Confusing momentum with kinetic energy. They use the same mass and velocity, but momentum is mv, in kg·m/s, with a direction, and kinetic energy is ½mv², in joules, with none.
Momentum FAQ
What is the formula for momentum?
p = mv, where m is the mass in kilograms and v the velocity in metres per second. The result is in kg·m/s. A 2 kg object moving at 3 m/s has a momentum of 6 kg·m/s.
How do you find the velocity from momentum?
Rearrange the formula: v = p / m. Divide the momentum by the mass. A 1500 kg car with 30,000 kg·m/s of momentum is moving at 20 m/s, which is 72 km/h.
What is impulse?
Impulse is the effect of a force acting over a time: J = FΔt, in newton seconds. It is also the change in momentum it causes, so a 1125 N kick lasting 0.01 s gives a ball 11.25 N·s of momentum.
What is the impulse-momentum theorem?
It says the impulse on an object equals its change in momentum: FΔt = m(v − v₀). It comes from Newton's second law, F = ma, with the acceleration written as the change in velocity over the time.
What are the units of momentum?
Kilogram metres per second, kg·m/s. The newton second, N·s, is the same unit, and it is the one usually used for impulse. The calculator also shows results in g·cm/s.
Can momentum be negative?
Yes. The sign shows the direction. If you call moving right positive, an object moving left has a negative velocity and a negative momentum. The size of the momentum is always positive.
How do you find the average force from a change in momentum?
Divide the change in momentum by the time it takes: F = m(v − v₀) / Δt. A 70 kg runner stopping from 10 m/s in 0.5 s needs an average force of 1400 N against the motion.
What is the difference between momentum and kinetic energy?
Momentum is mv and has a direction; kinetic energy is ½mv² and has none. Double the speed and the momentum doubles while the kinetic energy goes up four times.
