A 81.0 kg soccer player jumps vertically upwards and heads the 0.45 kg ball as it is descending vertically with a speed of 28.0 m/s. If the player was moving upward with a speed of 4.40 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?
To find the speed of the ball immediately after the collision, we can use the principles of conservation of momentum and the concept of elastic collision.
1. First, let's find the initial momentum of the system (player + ball) before the collision. The formula for momentum is:
momentum = mass × velocity
Momentum of the player before the collision:
momentum_player_initial = mass_player × velocity_player
where:
mass_player = 81.0 kg (mass of the soccer player)
velocity_player = 4.40 m/s (speed of the player before impact)
Therefore,
momentum_player_initial = 81.0 kg × 4.40 m/s
Momentum of the ball before the collision:
momentum_ball_initial = mass_ball × velocity_ball
where:
mass_ball = 0.45 kg (mass of the ball)
velocity_ball = -28.0 m/s (negative because the ball is moving downwards)
Therefore,
momentum_ball_initial = 0.45 kg × (-28.0 m/s)
2. The total initial momentum of the system is the sum of the individual momenta of the player and the ball:
total_initial_momentum = momentum_player_initial + momentum_ball_initial
3. According to the conservation of momentum, the total momentum before the collision should be equal to the total momentum after the collision (since no external forces act upon the system). Therefore,
total_initial_momentum = total_final_momentum
4. After the collision, the player and the ball move in opposite directions. Let's assume the final velocity of the player after the collision is v_player and the final velocity of the ball after the collision is v_ball. Hence,
total_final_momentum = mass_player × v_player + mass_ball × v_ball
5. Since the collision is elastic, kinetic energy is conserved. Therefore, we can also equate the initial kinetic energy to the final kinetic energy to solve for the velocities:
initial_kinetic_energy = final_kinetic_energy
(1/2) × mass_player × velocity_player^2 + (1/2) × mass_ball × velocity_ball^2 = (1/2) × mass_player × v_player^2 + (1/2) × mass_ball × v_ball^2
6. Now we have two equations (from step 3 and step 5) and two unknowns (v_player and v_ball). We can solve these equations simultaneously to find the final velocities.
7. Finally, after solving the equations, we can determine the speed of the ball immediately after the collision by taking the absolute value of v_ball.
Note: The negative sign in the velocity (-28.0 m/s) indicates that the ball is moving downward. Since the collision is elastic and the ball rebounds vertically upwards, the final velocity of the ball will be positive.
By following these steps and using the given values, you can calculate the speed of the ball immediately after the collision.