A player bounces a 0.46 kg soccer ball off her head, changing the velocity of the ball from i = (8.7 m/s) + (-3.7 m/s) to f = (5.1 m/s) + (4.2 m/s). The ball is in contact with the player's head for 6.7 ms.
To find the average force exerted by the player's head on the soccer ball, we can use Newton's second law of motion, which states that the force acting on an object is equal to the rate of change of its momentum.
The momentum of an object is given by the product of its mass and velocity.
Step 1: Find the initial momentum of the soccer ball.
The initial momentum of the soccer ball can be calculated by multiplying its mass with its initial velocity.
Mass of the soccer ball, m = 0.46 kg
Initial velocity of the soccer ball, vi = (8.7 m/s) + (-3.7 m/s) = 5.0 m/s
Initial momentum, p_initial = m * vi
Step 2: Find the final momentum of the soccer ball.
The final momentum of the soccer ball can be calculated by multiplying its mass with its final velocity.
Final velocity of the soccer ball, vf = (5.1 m/s) + (4.2 m/s) = 9.3 m/s
Final momentum, p_final = m * vf
Step 3: Calculate the change in momentum.
Change in momentum, Δp = p_final - p_initial
Step 4: Calculate the average force.
The average force can be found by dividing the change in momentum by the contact time.
Contact time, Δt = 6.7 ms = 6.7 * 10^(-3) s
Average force, F_average = Δp / Δt
Now, let's plug in the values and calculate the average force exerted by the player's head on the soccer ball.
F_average = (m * vf - m * vi) / Δt
F_average = (0.46 kg * 9.3 m/s - 0.46 kg * 5.0 m/s) / (6.7 * 10^(-3) s)
F_average = 4.3382 N
Therefore, the average force exerted by the player's head on the soccer ball is approximately 4.34 N.