Momentum Calculator

Momentum measures how hard something is to stop

p = mv combines the two properties that make a moving object difficult to arrest: how much of it there is, and how fast it is travelling. Both enter the product linearly, so a 6 kg mass at 5 m/s carries exactly the same momentum as a 3 kg mass at 10 m/s. The heavy slow object and the light fast one are equivalent as far as this quantity is concerned, and that even-handedness is what makes momentum the thing that survives an impact.

Momentum is a vector. It points where the object is going, and in any collision the vector sum before the impact equals the vector sum after it, provided no net external force acts on the colliding objects. Kinetic energy behaves quite differently. It is a scalar, it grows with the square of speed rather than the first power, and it is only conserved when the collision is elastic. Anything that crumples, heats up or makes a noise has lost kinetic energy while keeping every last unit of its momentum.

Worked example

A 3 kg mass moves at 10 m/s and is brought to rest in 0.5 s. These are the values loaded on screen, so you can change one and watch every figure below move with it.

• p = m v = 3 × 10 = 30 kg·m/s
• Impulse needed to stop it: 30 N·s, the same number
• F = p / t = 30 / 0.5 = 60 N
• KE = ½ m v² = 0.5 × 3 × 100 = 150 J

Note what happens if you double the speed to 20 m/s. Momentum doubles to 60 kg·m/s, but kinetic energy quadruples to 600 J. The two quantities do not track each other, which is why a slow lorry and a fast bullet are hazardous in quite different ways.

Stopping force is really about time

Impulse equals the change in momentum, and impulse is force multiplied by the time it acts for. In a crash the momentum you have to shed is already fixed by your mass and your speed, so the only quantity left to negotiate is the duration of the stop. Every safety feature in a car works that lever. Airbags, crumple zones and seatbelt pretensioners do not reduce the momentum to be lost. They stretch the interval over which it is lost.

The scaling is direct. Extending a stop from 10 milliseconds against a rigid dashboard to 100 milliseconds against an inflating bag cuts the average force by a factor of ten. Set the stopping time in the calculator to 0.05 s and the 60 N above becomes 600 N. The same reasoning explains why a wicketkeeper draws their hands back as they catch, and why a fall onto concrete injures where an identical fall onto a crash mat does not.

Common mistakes

• Treating momentum as a scalar. Two trolleys of equal mass closing head on at equal speed have a total momentum of zero, not double. Assign a positive direction, keep the signs, and only then add.
• Substituting kinetic energy for momentum. Momentum is mv and energy is ½mv². Conserving the wrong one is the single most common error in collision problems, because kinetic energy is conserved only in the idealised elastic case.
• Doubting that N·s equals kg·m/s. They are dimensionally identical and numerically interchangeable. Newton seconds are conventional for the impulse applied, kilogram metres per second for the momentum an object holds.
• Reading the stopping force as an instantaneous peak. The calculator returns the average force over the interval you supply. Real deceleration is far from uniform, and the peak force partway through a collision is comfortably higher than the average.

Converting units first? Use the mass, time, speed and momentum conversion tables.