Momentum calculator

How do I calculate momentum?

Momentum = mass × velocity. A 1,000 kg car at 20 m/s has 20,000 kg·m/s of momentum.

What units is momentum in?

Kilograms times meters per second (kg·m/s), which is the same as newton-seconds.

What is conservation of momentum?

In a collision with no outside forces, total momentum before equals total momentum after — the basis for working out collisions.

How is momentum different from kinetic energy?

Momentum grows in step with speed and has a direction; kinetic energy grows with the square of speed. The same car has 200,000 J of kinetic energy.

What is impulse?

Force × time — it equals the change in momentum. Stopping the car in 4 seconds takes an average force of 20,000 ÷ 4 = 5,000 N.

How do I find the force needed to stop a moving object?

Force times time equals the change in momentum, F·Δt = Δp (the impulse-momentum theorem, OpenStax University Physics, Volume 1, section 9.2, Eq. 9.7), so the average force is Δp ÷ Δt. The 3,000 lb car in the example below has 36,499 kg·m/s at 60 mph. Stopping it in 4 s takes an average force of 9,125 N (2,051 lbf); stopping it in 0.2 s takes 182,497 N, twenty times more. The same change in momentum over a longer time means a smaller force, which is the idea behind airbags (OpenStax Example 9.2).

Why can momentum be negative or cancel out?

Momentum is a vector with the direction of the velocity (OpenStax, section 9.1), so choose one direction as positive and give motion the other way a minus sign. Two 1,000 kg cars moving toward each other at 20 m/s each have momenta +20,000 and −20,000 kg·m/s, and the total is 0. The calculator works with the size of a single object’s momentum.

If two objects have the same momentum, do they have the same energy?

No. Kinetic energy can be written K = p² ÷ 2m, so for the same momentum the lighter object has more energy. Halve the car’s mass but keep its momentum (so its speed doubles) and the energy goes from 489.5 kJ to 979.0 kJ.

The formula and what it assumes

Momentum is mass times velocity, p = mv (OpenStax University Physics, Volume 1, section 9.1, Eq. 9.1). The SI unit is kg·m/s, which equals the newton-second, the unit of impulse. Momentum points the way the object moves; the calculator shows its size. The optional “Stop it in” time uses F = Δp ÷ Δt, which gives the average force for a constant slowdown; the peak force in a real collision can be higher.

Worked example: a 3,000 lb car at 60 mph

The mass is 3,000 × 0.45359237 = 1,360.8 kg and the speed is 60 × 0.44704 = 26.82 m/s. Then p = 1,360.8 × 26.82 = 36,499 kg·m/s, or 264,000 lb·ft/s in US units, where the pound is a mass. Slowing to a stop in 4 s means an average force of 36,499 ÷ 4 = 9,125 N. For comparison, OpenStax Example 9.5 finds a momentum change of 3.3 kg·m/s for a tennis ball of 0.057 kg served at 58 m/s.

Common mistakes

Confusing momentum with kinetic energy is the first: doubling the speed doubles the momentum but quadruples the energy. The second is the direction: in a collision, velocities opposite to your positive direction need a minus sign, or the conservation equation will not balance. The third is a weight given in newtons, which is not a mass: divide it by 9.80665 to get kilograms. Last, a speed in mph or km/h must be converted to m/s before multiplying by a mass in kilograms; the calculator does it when you pick the unit.

Sources: OpenStax University Physics, Volume 1, section 9.1 (linear momentum, Eq. 9.1), section 9.2 (impulse and collisions, Eq. 9.5–9.7; Examples 9.2 and 9.5) and section 7.2 (K = p² ÷ 2m); the pound, mile per hour and pound-force factors are from NIST SP 811, Appendix B.8. Related: collision calculator, kinetic energy calculator, force calculator and speed calculator.