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Classical Mechanics

Momentum & Collisions

What survives a collision, what doesn't, and why the centre of mass never notices.

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Momentum & Collisions

What survives a collision, what doesn't, and why the centre of mass never notices.

15 min2 checkpoints3 simulation tasks

Introduction

The forces in a collision are enormous, complicated and over in milliseconds, and you can predict the outcome without knowing any of them.

Predict

Two lumps of clay collide head-on and stick together, coming to a complete stop. Kinetic energy was clearly lost. What happened to the momentum?

Stop them dead

Try it

These carts stick together. The 3 kg cart comes in from the right at 2 m/s. Set the 1 kg cart's velocity to stop them dead, then run.

GoalMake the stuck pair stop dead.

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Momentum is mass times velocity, p⃗=mv⃗\vec{p} = m\vec{v}, and it has a direction. Add the momenta of everything in the collision: that total comes out exactly as it went in, whether the carts bounce apart or stick.

Where conservation comes from

Read the graph

The light cart's momentum goes from 9 to −7 kg·m/s in this collision. Run it. What does the heavy cart's momentum become?

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kg·m/s

Run the simulation first, then read the value off it.

The second law is a statement about momentum, Fnet=dp/dtF_{\text{net}} = dp/dt. By the third law the carts push on each other equally and oppositely, for the same time, so their momentum changes cancel:

Δp⃗1=−Δp⃗2⟹Δ(p⃗1+p⃗2)=0\Delta\vec{p}_1 = -\Delta\vec{p}_2 \quad\Longrightarrow\quad \Delta(\vec{p}_1 + \vec{p}_2) = 0
Practice

A kg cart moving at m/s hits a kg cart at rest, and they stick together. How fast do they move off?

m/s

Bounce, or stick

Try it

Two 1 kg carts meet head-on at 10 m/s each, carrying 100 J. Set the restitution so the collision loses half of that kinetic energy.

GoalLose half of the 100 J of kinetic energy.

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Practice

Set the restitution to 0.5. What is conserved and what is not?

What comes next

You stopped two carts dead by balancing their momenta, and made a collision shed half its energy while its momentum held. Everything so far has treated objects as points; next we let them spin, and a hoop and a sphere released from the same height turn out not to arrive together.

Ideas this lesson took on

Each of these is a common intuition the lesson put to the test. If one still feels right, the section that tests it is worth another look.

  • “Momentum is lost when a collision is inelastic”
  • “The heavier object exerts the larger force”
  • “Kinetic energy is always conserved”

Finished with this lesson?

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Research behind this lesson

  • Lawson & McDermott 1987: Students who could state both theorems often could not apply them, and frequently treated energy as a substance that gets used up rather than a quantity that is transferred.
  • Hestenes et al. 1992: A 29-item instrument, expanded to 30 in the 1995 revision, whose wrong answers are all documented student beliefs. It exposed how little conceptual understanding traditional courses were producing, and reoriented a generation of physics teaching.
  • Halloun & Hestenes 1985: Students arrive with a coherent, stable, Aristotelian-flavoured theory of motion. Conventional instruction leaves much of it intact: the formalism is learned alongside the intuition rather than in place of it.