A baseball player slides into third base with an initial speed of 7.75 m/s. If the coefficient of kinetic friction between the player and the ground is 0.45, how far does the player slide before coming to rest?
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To find the distance the baseball player slides before coming to rest, we can use the equation that relates the work done by friction to the change in kinetic energy.
The work done by friction is equal to the force of friction multiplied by the distance over which the player slides. The force of friction can be calculated using the coefficient of kinetic friction and the normal force exerted on the player.
First, let's calculate the force of friction. The normal force exerted on the player is equal to his weight, which can be calculated by multiplying his mass by the acceleration due to gravity (g ≈ 9.8 m/s^2). Let's assume the player's mass is 70 kg.
Weight = mass * acceleration due to gravity
Weight = 70 kg * 9.8 m/s^2 = 686 N
Next, let's calculate the force of friction.
Force of friction = coefficient of kinetic friction * normal force
Force of friction = 0.45 * 686 N = 308.7 N
Now, let's calculate the work done by friction. The work done by friction is equal to the force of friction multiplied by the distance over which the player slides. We are trying to find the distance, so let's call it d.
The work done by friction = force of friction * distance
The work done by friction = 308.7 N * d
The work done by friction is equal to the change in kinetic energy, which is given by the initial kinetic energy (which is 1/2 * mass * initial velocity^2) minus the final kinetic energy (which is 0 since the player comes to rest).
The work done by friction = change in kinetic energy
Now we can set up the equation:
308.7 N * d = 0.5 * 70 kg * (7.75 m/s)^2 - 0
Let's solve for d:
d = (0.5 * 70 kg * (7.75 m/s)^2) / 308.7 N
Calculating this equation, we get:
d ≈ 3.96 meters
Therefore, the baseball player slides approximately 3.96 meters before coming to rest.