Use two conservation laws in an elastic collision

1. Orient: Use two conservation laws in an elastic collision

An elastic collision preserves the total kinetic energy of the colliding system. If external impulse is negligible, it also preserves total momentum. Individual objects can exchange both momentum and kinetic energy; neither conservation law says each object keeps its own initial value.

2. Explain and derive

For one-dimensional masses m 1,m 2 and velocities u 1,u 2 before collision, momentum gives m 1 u 1+m 2 u 2=m 1 v 1+m 2 v 2. Kinetic-energy conservation gives m 1 u 1²+m 2 u 2²=m 1 v 1²+m 2 v 2² after canceling½. Factoring the differences of squares and combining the nontrivial collision solution yields v 2−v 1=u 1−u 2: relative approach and separation speeds match.

Solving the linear momentum equation with that relative-speed relation gives v 1=[(m 1−m 2)u 1+2 m 2 u 2]/(m 1+m 2) and v 2=[2 m 1 u 1+(m 2−m 1)u 2]/(m 1+m 2). Equal masses exchange velocities. That convenient special case should not be applied to unequal masses.

The model stipulates elasticity rather than measuring it. A real collision that does not stick may still lose kinetic energy to deformation, sound or internal motion. To classify it, compare before-and-after momentum and energy with uncertainty and external-interaction estimates, not merely whether the objects bounce.

3. Work an example

With masses 2 and 3 kg and incoming velocities+4 and−1 m/s, the formulas give v 1=−2 m/s and v 2=+3 m/s. Momentum before is 8−3=5 and after is−4+9=5 kg m/s. Energy before is 16+1.5=17.5 J and after is 4+13.5=17.5 J. Relative separation speed is 5 m/s.

4. Predict and investigate

Predict first: If the masses are equal, which velocity does each object receive? Does that mean the same rule holds when only one mass doubles?

Compare equal masses with(2,3) and(3,2) kg. Reconstruct total momentum and kinetic energy from the individual final velocities. Check the 5 m/s relative separation and explain why individual energies can differ from their initial values. Retain settings and raw evidence in the lesson notebook.

5. Transfer and explain

Try a new practice case

A 1 kg moving object at 6 m/s collides elastically with an equal stationary mass in one dimension. Find the second object’s final velocity.

Your numerical result (m/s) Check number

Reveal worked check (practice, not mastery)

This extra practice is not saved and does not award mastery. Record your first answer, explanation and whether you revealed help in the lesson notebook. Use the lesson’s saved question separately.

6. Reflect and retain evidence

What evidence would rule out a perfectly elastic model even if measured total momentum were compatible with conservation?

Retain the first answer, signed component sums, scalar energy totals, bounds, assumptions and one changed-case prediction. Public worked checks are practice, not independent mastery evidence. Synthetic readings are not a performed physical experiment.

Framework and source notes

Original teaching prose and numerical cases. College Board AP Physics 1 CED, Unit 4 topic mapping: 4.3; 4.4. Checked 30 October 2026. Two-dimensional cases use component setup and simple separate relations, not a general simultaneous-angle solver. This is a framework cross-reference, not AP authorization.