Two-Dimensional Collision Questions

Two-dimensional collision questions combine everything in this topic. A reliable approach:
1. Sketch all velocities (with angles) before and after.
2. Resolve every momentum along x and y.
3. Conserve momentum in x and in y separately.
4. Solve, then recombine with Pythagoras and tan.
5. If asked, compare total kinetic energy before and after to test for an elastic collision.Worked example: a 0.20 kg ball moving at 5.0 m s^-1 strikes an identical stationary ball and moves off at 4.0 m s^-1 at 37° to its original direction. Find the second ball's velocity; is the collision elastic?
momentum before: 0.20 × 5.0 = 1.0 kg m s^-1 along x
first ball after: 0.80 kg m s^-1, so p_x = 0.80 cos 37° = 0.64 and p_y = 0.80 sin 37° = 0.48
x: 1.0 = 0.64 + p_x, so p_x = 0.36 kg m s^-1
y: 0 = 0.48 + p_y, so p_y = –0.48 kg m s^-1
p = √(0.36² + 0.48²) = 0.60 kg m s^-1, so v = 0.60 / 0.20 = 3.0 m s^-1 at tan^-1(0.48 / 0.36) = 53° on the other side of the x-axis.
E_k before = ½ × 0.20 × 5.0² = 2.5 J; after = 1.6 J + 0.9 J = 2.5 J, so the collision is elastic.Notice the balls separate at 37° + 53° = 90°. After an elastic glancing collision between a moving ball and an identical stationary ball, they always move off at 90° to each other.