Impulse and Momentum Simulator

What Is the Impulse and Momentum Simulator?

Impulse: Why a Longer Stop Hurts Less

Your car is doing 30 and you have to stop. Whether you brake gently or slam into a wall, your change in momentum is identical — the same Δp either way. What differs is the time you take to lose it. That is the whole idea behind impulse: J = F·Δt = Δp. Momentum p = m·v is what you carry; impulse is what you must deliver to change it, and it equals that momentum change exactly (the impulse-momentum theorem, J = m·Δv).

Now rearrange for force: F = Δp/Δt. Force is the rate at which momentum changes, so the same Δp can arrive as a huge force over a blink, or a gentle force spread across a long moment. Drag the Time slider up and watch the force readout fall while the impulse stays fixed. That single relationship explains airbags, crumple zones, and bending your knees on landing — each one stretches Δt to shrink F. Push Time down and the force spikes: the sudden stop is the one that breaks bones.

In an isolated collision the total momentum is conserved, and the two bodies trade equal, opposite impulses. Set Mass A, Velocity A, Mass B and Velocity B, then read each p, J and F side by side. For a numeric check drop values into the impulse calculator, or explore two carts meeting in the momentum lab and the rest of the interactive physics collection.

Where the ideal impulse-momentum picture breaks down

The two collision buttons are not two options but the two ends of a scale: the coefficient of restitution, 1 for perfectly elastic, 0 for sticking. Steel bearings come in near 0.9, a tennis ball on a hard court around 0.75, modelling clay close to 0. Send 2 kg at 4 m/s into a stationary 1 kg and the lab offers either 1.33 and 5.33 m/s, or 2.67 m/s for both. Real carts land between those answers, and this lab has no dial for it.

Impulse mode hides a second idealisation in plain sight. The shaded block is a rectangle — constant force for the whole contact. A real contact force starts at zero, peaks as the surfaces compress, then falls back. Only the area is trustworthy, so the readout prints the average force, never the worst instant. At Δp = 20 kg·m/s and Δt = 0.20 s it reads 100 N; shape that impulse as a half sine wave and the peak is π/2 times higher, 157 N, or triangular and it is 200 N. Design a helmet liner to the average and you under-size it twofold.

Nothing here deforms, either. The carts swap velocities in a single frame, as though contact took no time; in a real impact the bumper is already moving at a different speed from the passenger. That deformation is where the missing energy goes: couple 10 kg at 2 m/s to a stationary 10 kg and momentum holds at 20.00 kg·m/s while kinetic energy falls from 20.0 J to 10.0 J. Those 10 J are the bent metal, the warmed buffer and the bang you hear.

Finally, the track is a straight line and every hit is square. Strike off-centre and the bodies spin: momentum still balances, but energy goes into rotation and the straight-line speeds fall below what this lab predicts. Momentum also balances only for an isolated system, and since the lab computes both totals from the same law they always agree — while a real air track leaks a little to friction between the photogates.

Where this is actually used

Seat belts and airbags. In Impulse mode, set Δp to 50 kg·m/s and Δt to 0.02 s: the force reads 2500 N. Drag Δt to 1.00 s and the identical momentum change needs 50 N, a fifty-fold cut bought with time alone. At human scale a 75 kg occupant at 14 m/s (about 50 km/h) carries 1050 kg·m/s: stopped in 0.05 s that averages 21,000 N; spread over 0.3 s by belt and airbag, 3,500 N.

Railway couplers. Two wagons coupling is the sticking case, equal masses, one at rest. Run Mass A 10 kg at 2 m/s into Mass B 10 kg at 0 m/s, Inelastic: both leave at 1.00 m/s, momentum reads 20.00 kg·m/s before and after, and kinetic energy halves from 20.0 J to 10.0 J. That half holds for any equal-mass coupling from rest — about 17 kJ for two 30-tonne wagons closing at 1.5 m/s, which is why couplers sit behind a sprung draft gear.

Choosing a reactor moderator. A head-on elastic hit hands over the fraction 4·mA·mB/(mA+mB)2 of the incoming energy. Set both masses to 1 kg with A at 5 m/s: A stops dead at 0.00 m/s and B carries away the whole 12.5 J. Make B 10 kg and send A in at 10 m/s: A rebounds at -8.18 m/s keeping two thirds of the 50.0 J, handing over 33%. That is the argument for moderating neutrons with light nuclei — water or graphite, not lead.