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?
Details and assumptions
There is no friction between the rollerblades and the surface, and Mary can slide freely.
Mary's first throw is from the rest.
The ball bounces elastically from the wall.
Mary throws the ball such that it always has the same speed with respect to the ground.
Neglect any gravitational influences.
To determine the maximum ratio of Mary's mass to the ball's mass in order for this situation to occur, we need to analyze the physics involved.
Let's break it down step by step:
1. The ball is thrown towards the wall: When Mary throws the ball towards the wall, it exerts a force on the ball, propelling it forward.
2. The ball bounces off the wall: When the ball hits the wall, it undergoes an elastic collision. This means that its speed remains the same, but its direction is reversed.
3. Mary tries to catch the ball: After the ball bounces off the wall, Mary tries to catch it. However, since she is wearing rollerblades, she is unable to stop herself from sliding backwards.
The key to solving this problem lies in understanding the conservation of momentum. According to Newton's third law of motion, every action has an equal and opposite reaction. When Mary throws the ball towards the wall, she exerts a force on it, and the ball exerts an equal and opposite force on her. This force causes Mary to slide in the opposite direction.
To reach a maximum ratio of Mary's mass to the ball's mass, we need to find the point at which Mary's momentum after throwing the ball is exactly opposite to the ball's momentum after bouncing off the wall. This ensures that the forces cancel out and Mary's momentum matches the ball's momentum.
Mathematically, we can express this as:
Mary's momentum after throwing the ball = Ball's momentum after bouncing off the wall
Since momentum is defined as the product of mass and velocity (momentum = mass x velocity), we can rewrite this equation as:
Mary's mass x Mary's velocity = Ball's mass x Ball's velocity
Since we know that the ball's velocity remains the same after bouncing off the wall, Mary's velocity must also remain the same. Let's denote the ratio of Mary's mass to the ball's mass as "r." We can rewrite the equation as:
r x Mary's mass x Mary's velocity = Mary's mass x Ball's velocity
We can simplify this equation by canceling out Mary's mass:
r x Mary's velocity = Ball's velocity
Now, we can solve for the maximum value of "r" by considering the relative velocities. Since Mary's velocity must be in the opposite direction to the ball's velocity and have the same magnitude, the maximum value of "r" occurs when:
r = |Mary's velocity| / |Ball's velocity|
In other words, the maximum ratio of Mary's mass to the ball's mass is equal to the magnitude of Mary's velocity divided by the magnitude of the ball's velocity.
To find this ratio, you would need to know the specific values of Mary's velocity and the ball's velocity. Without these values, it is not possible to determine the exact maximum ratio.