Adventurous Mary tries a new game: she puts on her rollerblades and throws the ball towards the wall, lets it bounce off, catches it again, throws it right back, and so on. She notices a strange thing: after the second throw, she is unable to catch the ball anymore since the ball never reaches her. What is the maximum ratio of Mary's mass to the ball's mass in order for this to happen?
To determine the maximum ratio of Mary's mass to the ball's mass in order for her to be unable to catch the ball after the second throw, we need to consider the principle of conservation of momentum.
When Mary throws the ball towards the wall, she imparts momentum to it. Upon rebound, the ball changes direction and travels back towards Mary. If Mary wants to catch the ball, she needs to counteract its backward momentum with her own forward momentum by moving towards the ball. In doing so, she also experiences a change in momentum.
For Mary to be unable to catch the ball after the second throw, the backward momentum of the ball needs to be greater than the forward momentum she can generate by moving towards the ball. So, we can set up an equation to represent this:
Ball's momentum after rebound = Mary's momentum after moving towards the ball
The momentum of an object is calculated as the product of its mass and velocity. Let's assume that Mary's initial velocity throwing the ball and rebound velocity of the ball are the same, denoted by 'v'. Also, let's assume Mary's velocity towards the ball after the second throw is 'v_m' and the ball's velocity towards Mary is 'v_b'. The equation can then be written as:
(-m_ball * v) = (m_mary * v_m) + (m_ball * v_b)
Where:
m_ball = mass of the ball
m_mary = mass of Mary
Since the ball's velocity immediately after the rebound is the same magnitude but opposite in direction to its velocity initially thrown by Mary, we have v_b = -v.
Substituting this into the equation, we get:
-m_ball * v = m_mary * v_m - m_ball * v
Rearranging the equation, we have:
m_ball * v = m_mary * v_m + m_ball * v
Simplifying further, we get:
m_ball * 2v = m_mary * v_m
Now, to determine the maximum ratio of Mary's mass to the ball's mass, we need to find the point where she can no longer catch the ball. In this situation, Mary is unable to generate any forward momentum (v_m = 0) to counteract the ball's backward momentum. Hence, the equation becomes:
m_ball * 2v = 0
Since we want to find the maximum ratio, we assume that the ball's mass is nonzero, so m_ball ≠ 0. From this equation, we can see that the maximum ratio of Mary's mass to the ball's mass occurs when her mass is infinity (m_mary = ∞) and the ball's mass is any non-zero value.
Therefore, the maximum ratio of Mary's mass to the ball's mass for her to be unable to catch the ball after the second throw is ∞ : 1 (infinity to one).