Momentum & Impulse

Try this thought experiment: catch a raw egg by pulling your hands backward as it lands. The egg survives. Now catch the same egg with stiff hands, palms facing forward. The egg shatters.

In both cases, the egg’s momentum changes by the same amount (it goes from “falling” to “stopped”). What changed is the time over which that change happened. The soft, backwards-pulling catch took longer, so the force on the egg was smaller. The stiff catch was abrupt — same momentum change packed into a tiny time, which means a huge force, which means a broken egg.

That trade-off — same change in momentum, different force depending on how long the change takes — is the impulse-momentum theorem, and it shows up everywhere on the MCAT: car crashes, airbags, padded helmets, bending your knees when you land from a jump, rocket propulsion. Master it once and you’ve explained half of safety engineering.

Linear Momentum

Momentum is sometimes called “inertia in motion.” It’s a measure of how hard it is to stop something that’s already moving. A slow-moving freight train can have the same momentum as a fast-moving car, because the train’s huge mass makes up for its low speed. A stationary object ($v = 0$) has zero momentum no matter how massive it is.

Conservation of Momentum

In an isolated system (no net external force), the total momentum before any event equals the total momentum after:

What counts as “isolated”? External forces (friction with the ground, gravity, etc.) must be negligible during the interaction. Most collision problems treat the collision as so brief that external forces have no time to change momentum significantly — even if those forces are present at other times. Two billiard balls on a felt table aren’t a perfectly isolated system (friction is there), but during the millisecond of the actual collision, friction’s effect is tiny, so we treat the collision as if it were isolated.

Impulse

Impulse ($J$) is the change in momentum, and it equals the average net force multiplied by the time the force acts:

The whole punch of this equation is the trade-off between $F$ and $\Delta t$. For a fixed momentum change ($\Delta p$, set by the situation), making $\Delta t$ bigger makes $F$ smaller. That’s the entire principle behind every safety device ever invented.

Impulse on a Force-Time Graph

Impulse equals the area under a force-time graph. If the force is constant, the area is just a rectangle ($F \times \Delta t$). If the force varies (most real impacts), the impulse is the area under whatever curve the force traces. The peak force can be much larger than the average force — and it’s the peak that breaks bones, not the average.

Real-World Applications

Solving Momentum Problems

Same recipe every time:

1. Define the system. Which objects are included?
2. Check for isolation. Are net external forces negligible during the interaction?
3. Draw “before” and “after” diagrams with velocity vectors.
4. Pick a positive direction (and stick with it).
5. Apply conservation of momentum: total $p$ before = total $p$ after.
6. Solve for the unknown.

Worked example. A 5 kg cart moving at 4 m/s right collides with a stationary 3 kg cart. After the collision, the 5 kg cart moves at 1 m/s right. What is the velocity of the 3 kg cart?

• Before: $p_{total} = 5(4) + 3(0) = 20$ kg·m/s.
• After: $20 = 5(1) + 3v_2 \Rightarrow 15 = 3v_2 \Rightarrow v_2 = 5$ m/s (to the right).

So the smaller cart now moves at 5 m/s right — faster than the original cart was moving. That’s allowed because momentum, not speed, is conserved. The smaller cart has less mass, so it can carry the missing momentum at a higher speed.