Momentum and collisions

A ping-pong ball and a bowling ball can roll at the exact same speed, yet one you stop with a fingertip and the other breaks your foot. Speed alone does not tell you how hard something is to stop. There is a second ingredient, and crashes are built entirely out of it.

Think about catching things. A tennis ball tossed to you is nothing. A medicine ball thrown at the same speed knocks you back a step. Now imagine the tennis ball fired from a machine at a hundred miles an hour. Suddenly it stings. Whatever it is that makes a thing hard to stop, it grows with both how heavy the thing is and how fast it moves. Physics gives that combined quantity a name, and it turns out to be the single most reliable thing to track through any collision.

First, predict

Two identical shopping carts roll straight at each other, same speed, and lock together on impact. The instant after they meet, before anything else happens, what does the combined wreck do?

Play with the crash first

Before any equation, get a feel for it. Set the two masses and speeds, then drag the elasticity slider from a perfectly bouncy hit down to a perfectly sticky one, and Launch. Watch the two readouts under the track as the pucks meet. One of them never changes. The other only survives when the crash is perfectly bouncy.

Try the predict scenario: give both pucks equal mass and opposite speeds, set elasticity to zero, and Launch. They stop. Now nudge one mass up, and the wreck drifts toward the heavier side. Whatever is being kept constant clearly cares about direction, and about mass and speed together.

Momentum: mass times velocity

That conserved thing is momentum. Heavy or fast both raise it, and it points the way the object moves.

This is not a new law. It falls out of the F = ma we already have. Acceleration is just how fast velocity changes, so:

Read that last step slowly, because it is the whole lesson in one line (DERIVED, not asserted). A force is the rate momentum changes. Rearranged, a force acting for a stretch of time delivers a chunk of momentum, and that chunk has its own name.

Impulse: force multiplied by time

Multiply both sides by the time the force acts and you get impulse, the total change in momentum a push produces.^[1]

This is the key to every safety device you own. When a car stops in a crash, its change in momentum is fixed: it has to go from full speed to zero, whatever happens. A crumple zone cannot change that $\Delta \mathbf{p}$. What it does is stretch out the $\Delta t$. Same momentum change, spread over more time, means a smaller force, because $\mathbf{F}=\Delta \mathbf{p}/\Delta t$.^[2] Airbags, bike helmets, and the bend in your knees when you land are all the same trick: make the stop take longer.

Why the total never changes

Now the conservation you watched in the widget. When two things collide, the interaction pair rule says they push on each other with equal and opposite force, for exactly the same length of time. Equal and opposite forces, equal time, means equal and opposite impulses:

So the total momentum of the pair is the same before and after (DERIVED from the interaction pair). No outside push acts along the track, and the two internal pushes cancel, so the sum has nothing that can change it. This is why the readout in the widget held steady no matter how you set the slider.

Three kinds of collision

The elasticity slider walked you across a spectrum, with two clean endpoints and an interesting third case.^[2]

A perfectly elastic collision hit keeps both momentum and kinetic energy: think two hard billiard balls, or the bouncy end of the slider. An inelastic collision hit keeps momentum but converts some motion energy into heat and damage; sticking together is the extreme. And an explosion runs it backwards: it starts with zero momentum and blows apart into pieces whose momenta still sum to zero, while stored chemical energy increases the kinetic energy. A rocket is exactly this: it throws gas one way and gains momentum the other, total still balancing to zero.

The same thing in code

A one-dimensional collision is a five-line function, and it is the exact update the widget runs each time the pucks touch. Notice the assertion at the end: momentum before equals momentum after, always.

def collide(mA, vA, mB, vB, e):
    p = mA * vA + mB * vB          # total momentum, before
    rel = vA - vB                  # closing speed
    vA2 = (p - mB * e * rel) / (mA + mB)
    vB2 = (p + mA * e * rel) / (mA + mB)
    assert abs((mA*vA2 + mB*vB2) - p) < 1e-9   # momentum unchanged
    return vA2, vB2

# 2 kg at +3 m/s hits a still 4 kg puck, perfectly sticky (e = 0)
print(collide(2, 3, 4, 0, e=0))    # -> both move at +1.0 m/s together

Check yourself