The Colliding Balls: A ball is released from rest at a height h above the ground. The ball collides with the ground and bounces up at 75% of the impact speed the ball had with the ground. The collision with the ground is nearly instantaneously. A second ball is released above the first ball from the same height the instant the first ball loses contact with the ground.
a.What is the time when the two balls collide?
b.What is the height in terms of h where the collision occurs?
To find the answers to both parts of this question, we can break it down into the following steps:
Step 1: Analyze the motion of the first ball.
- The first ball is dropped from rest at a height h, so we can use the equation for free fall:
h = (1/2)gt^2
where g is the acceleration due to gravity and t is the time taken to fall from height h to the ground.
Step 2: Determine the speed of the first ball just before the collision.
- Since the ball bounces up at 75% of the impact speed it had with the ground, we can calculate the velocity just before it hits the ground:
v_impact = sqrt(2gh)
where v_impact is the impact velocity just before the collision.
Step 3: Calculate the time taken for the first ball to collide with the ground.
- We can find the time taken for the first ball to reach the ground using the equation from Step 1:
h = (1/2)gt^2
Solve this equation for t to find the time taken for the first ball to collide with the ground.
Step 4: Determine the time when the two balls collide.
- Since the second ball is released from the same height as the first ball the moment it loses contact with the ground, the time taken for the second ball to reach the ground will be the same as the time taken for the first ball to reach the ground.
Step 5: Calculate the height where the collision occurs.
- Since the second ball is released at the same height as the first ball, and the time taken for the second ball to reach the ground is the same as the time taken for the first ball to reach the ground, the height where the collision occurs will also be h.
By following these steps, you can find the time and height where the two balls collide in terms of the initial height, h.