A 50.0 kg soccer player jumps vertically upwards and heads the 0.45 kg ball as it is descending vertically with a speed of 27.0 m/s. If the player was moving upward with a speed of 2.80 m/s just before impact. (a) What will be the speed of the ball immediately after the collision if the ball rebounds vertically upwards and the collision is elastic?
(Note that the force of gravity may be ignored during the brief collision time.)
To find the speed of the ball immediately after the collision, we can use the principle of conservation of momentum.
Step 1: Calculate the initial momentum of the player and the ball before the collision.
The initial momentum of the player is given by:
Momentum_player = mass_player * velocity_player
Momentum_player = 50.0 kg * 2.80 m/s = 140 kg·m/s
The initial momentum of the ball is given by:
Momentum_ball = mass_ball * velocity_ball
Momentum_ball = 0.45 kg * (-27.0 m/s) = -12.15 kg·m/s
Step 2: Calculate the final momentum of the player and the ball after the collision.
Since the collision is elastic and the ball rebounds vertically upwards, the final momentum of the player will be the same as the initial momentum:
Momentum_player_final = 140 kg·m/s
The final momentum of the ball can be calculated using the conservation of momentum:
Momentum_ball_final = -Momentum_player_final
Step 3: Calculate the speed of the ball immediately after the collision.
The final momentum of the ball is given by:
Momentum_ball_final = mass_ball * velocity_ball_final
Rearranging the equation, we get:
velocity_ball_final = Momentum_ball_final / mass_ball
Substituting the values, we find:
velocity_ball_final = -Momentum_player_final / mass_ball
velocity_ball_final = -140 kg·m/s / 0.45 kg
Using a calculator, we find:
velocity_ball_final = -311.11 m/s
Since the negative sign indicates the direction, we can disregard it for the speed. Therefore, the speed of the ball immediately after the collision is approximately 311.11 m/s going upward.
To solve this problem, we can use the principle of conservation of momentum and the principles of one-dimensional elastic collisions. Here are the steps to find the speed of the ball immediately after the collision:
1. Calculate the initial momentum of the player and the ball separately before the collision.
- The initial momentum of the player (m1) can be calculated as: m1 = mass of the player * velocity of the player = 50.0 kg * 2.80 m/s.
- The initial momentum of the ball (m2) can be calculated as: m2 = mass of the ball * velocity of the ball = 0.45 kg * 27.0 m/s.
2. Calculate the final momentum of the player and the ball separately after the collision using the conservation of momentum.
- The final momentum of the player (m1') can be calculated as: m1' = m1.
- The final momentum of the ball (m2') can be calculated as: m2' = -m2 (since the ball is rebounding vertically upwards).
3. Use the final momentum of the ball to calculate its final velocity.
- The final velocity of the ball (v2') can be calculated as: v2' = m2' / mass of the ball.
Here, m2' is the final momentum of the ball obtained in the previous step, and the mass of the ball is given as 0.45 kg.
4. Plug in the values and calculate the final velocity of the ball.
- Substitute the values in the formula: v2' = m2' / mass of the ball.
By following these steps and solving the equations, you can find the speed of the ball immediately after the collision.