A 55-kg soccer player jumps vertically upwards and heads the 0.45-kg ball as it is descending vertically with a speed of 30 m/s.
(a) If the player was moving upward with a speed of 4.0 m/s just before impact, what will be the speed of the ball immediately after the collision if the ball rebounds vertically upwards and the collision is elastic?
1 m/s
(b) If the ball is in contact with the player's head for 23 ms, what is the average acceleration of the ball? (Note that the force of gravity may be ignored during the brief collision time.)
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(a) To find the speed of the ball immediately after the collision, we can use the law of conservation of momentum.
The momentum before the collision is equal to the momentum after the collision.
The initial momentum of the ball is given by:
P_initial = mass_ball * velocity_ball_initial = 0.45 kg * (-30 m/s) = -13.5 kg*m/s
The initial momentum of the player is given by:
P_initial = mass_player * velocity_player_initial = 55 kg * 4.0 m/s = 220 kg*m/s
Since the collision is elastic, there is no loss of kinetic energy. Therefore, the total momentum before the collision is equal to the total momentum after the collision:
P_initial = P_final
(-13.5 kg*m/s) + (220 kg*m/s) = (0.45 kg * velocity_ball_final) + (55 kg * velocity_player_final)
Simplifying the equation, we get:
206.5 kg*m/s = (0.45 kg * velocity_ball_final) + (55 kg * velocity_player_final)
To find the velocity of the ball immediately after the collision, we need to isolate the velocity_ball_final term:
0.45 kg * velocity_ball_final = 206.5 kg*m/s - (55 kg * velocity_player_final)
velocity_ball_final = (206.5 kg*m/s - (55 kg * velocity_player_final)) / 0.45 kg
Plugging in the given values:
velocity_ball_final = (206.5 kg*m/s - (55 kg * 4.0 m/s)) / 0.45 kg
Solving this equation, we find:
velocity_ball_final = 1 m/s
So, the speed of the ball immediately after the collision will be 1 m/s.
(b) To find the average acceleration of the ball during the collision, we can use the equation:
acceleration = change in velocity / time
The change in velocity can be calculated using the equation from part (a):
change in velocity = velocity_ball_final - velocity_ball_initial
Since the ball rebounds vertically upwards, the final velocity will be positive:
velocity_ball_final = 1 m/s
velocity_ball_initial = -30 m/s
change in velocity = 1 m/s - (-30 m/s) = 31 m/s
The time is given as 23 ms, which needs to be converted to seconds:
time = 23 ms / 1000 = 0.023 s
Now we can calculate the average acceleration:
average acceleration = change in velocity / time = 31 m/s / 0.023 s
Solving this equation, we find:
average acceleration = 1347.83 m/s^2
So, the average acceleration of the ball during the collision is approximately 1347.83 m/s^2.