Superballs

Superballs have nearly elastic collisions with both hard floors and other superballs. A small superball (mass \(m_s\)) is dropped directly on top of a large superball of mass \(m_l\), both falling from rest from a height \(h\). After an initial collision with the floor, the large superball heads directly upwards at speed \(v\), where it collides elastically with the smaller ball, also moving at speed \(v\) downward.

Part 1

If the large ball has speed \(0\) following the elastic collision, what is the final speed of the smaller ball, \(v_f\)?

Use conservation of momentum.

Write your answer in terms of the mass of the larger ball \(m_l\), the mass of the smaller ball \(m_s\), and velocity \(v\).

Use the following table as a reference for each variable. Note that it may not be necessary to use every variable.

For	Use
\(m_l\)	m_l
\(m_s\)	m_s
\(v\)	v

Answer Section

Part 2

In order for this to be an elastic collision, what must the ratio of the masses be?

Answer Section

Part 3

Following this collision, what maximal height will the small ball reach?

Express your answer in terms of \(h\), the height from which both balls were dropped.

For	Use
\(h\)	h

Answer Section

Part 4

If we replaced the top ball with a “sad” ball, which undergoes a perfectly inelastic collision with the returning “happy” ball, to what height would the “sad” ball return following the collision in terms of \(h\), the height from which both balls were dropped? (For this question, assume that \(\frac{m_l}{m_s} = \) 3).

For	Use
\(h\)	h

Answer Section

Attribution

Problem is licensed under the CC-BY-NC-SA 4.0 license.