Original lab · ideal rigid bodies · one dimension

What survives a collision?

Two carts can exchange velocity without losing total momentum. Kinetic energy is a different story. Change the inputs and compare the instant before contact with the instant after.

Try the cart inputs

Only straight-line translation is modeled: cart A starts left of B on a frictionless track, with no external impulse. No rotation, two-dimensional impacts, deformation history or collision duration is simulated.

Change a cart. Read the consequence.

Positive velocity means right; negative means left. Optional prediction: with equal masses and restitution 1, which cart will leave with A’s starting velocity?

Results update immediately. Use arrow keys on each slider. Reset restores only these five inputs; nothing is saved.

Cart A · starts on the left
Cart B · starts on the right

0: no relative separation speed after impact. 1: elastic, with no kinetic-energy loss.

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Two constraints determine the result

Let u be starting velocity, v be final velocity and m be mass. Equal and opposite impulses conserve total momentum. Restitution sets the ratio of separation speed to approach speed.

mA uA + mB uB = mA vA + mB vB
vB − vA = e (uA − uB)

When uA > uB, the gap closes. Solving these two equations gives the impulse J and the outgoing velocities:

J = (1 + e)(uA − uB) / (1/mA + 1/mB)
vA = uA − J/mA
vB = uB + J/mB

When uA ≤ uB the gap does not close: this lab applies no impulse and leaves both velocities unchanged. Restitution is then irrelevant, not evidence of an elastic impact.

Kinetic energy is K = (mA uA² + mB uB²)/2 before impact. The loss for an approaching pair is ½ [mA mB/(mA + mB)] (1 − e²)(uA − uB)². At e = 1 it is zero. At e = 0 the carts share a velocity, but can still carry kinetic energy together. Lost translational kinetic energy represents unmodeled internal energy, sound or deformation; total energy is not destroyed.

Momentum is not kinetic energy

Try a heavier B, then reduce restitution to zero. Compare the unchanged total momentum with the reduced kinetic energy. Finally set A moving left and B moving right: there is no future collision to resolve.

This independently written lesson replaces an earlier archived article at the same URL. Its four documented section links remain entry points, but it does not reproduce that article’s multidimensional demonstrations. Model and implementation notes.