# 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

## Part 2#

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

## 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

## 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 $$\frac{m_l}{m_s} =$$ 4).

For

Use

$$h$$

h