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, let's analyze the physics behind the situation.
When Mary throws the ball against the wall, it will bounce off with a certain velocity V. According to the law of conservation of momentum, the ball and Mary's combined system must have the same total momentum before and after the collision.
Considering the system consisting of Mary and the ball, we can calculate the momentum before the collision as the product of the combined mass (m_M+ m_b) and the initial velocity of the ball, which is V.
Momentum before collision = (m_M+ m_b) * V
After the collision, Mary will be moving in the opposite direction with a certain velocity V_m (assuming she does not slip or change her position significantly). Meanwhile, the ball will have a new velocity V_b.
Momentum after collision = m_M * V_m + m_b * V_b
Since Mary catches the ball on her subsequent throws, it means that the ball has zero change in momentum, implying that the velocity of the ball after bouncing is equal to its velocity before bouncing.
V_b = -V
Now, let's set up an equation based on the conservation of momentum:
(m_M+ m_b) * V = m_M * V_m + m_b * (-V)
Simplifying the equation:
(m_M+ m_b) * V = m_M * V_m - m_b * V
Rearranging the equation:
m_M * V_m = (m_M+ m_b + m_b) * V
m_M * V_m = (m_M+ 2m_b) * V
To solve for the maximum ratio of Mary's mass to the ball's mass, we can divide both sides of the above equation by m_b * V_m:
m_M / m_b = (m_M+ 2m_b) / V
Now, we need to find the point where Mary cannot catch the ball, which means the ball never reaches her. This happens when the velocity of Mary, V_m, is zero.
m_M / m_b = (m_M+ 2m_b) / 0
However, dividing by zero is undefined. Therefore, if Mary is unable to catch the ball after the second throw, the maximum ratio of Mary's mass to the ball's mass does not exist.
In conclusion, there is no maximum ratio of Mary's mass to the ball's mass for her to be unable to catch the ball after the second throw.