A 85.0 kg base runner begins his slide into second base when he is moving at a speed of 2.10 m/s. The coefficient of friction between his clothes and Earth is 0.570. He slides so that his speed is zero just as he reaches the base. (a) How much mechanical energy is lost due to friction acting on the runner? b) how far does he slide?
To answer question (a), we need to calculate the mechanical energy lost due to friction acting on the runner. We can use the work-energy principle to solve this problem.
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by friction is equal to the change in kinetic energy of the runner.
The change in kinetic energy can be calculated using the formula:
ΔKE = KE_final - KE_initial
where KE_final is the final kinetic energy, and KE_initial is the initial kinetic energy.
The initial kinetic energy can be calculated using the formula:
KE_initial = 0.5 * mass * velocity^2
Given that the mass of the runner is 85.0 kg and the initial velocity is 2.10 m/s, we can calculate KE_initial:
KE_initial = 0.5 * 85.0 kg * (2.10 m/s)^2
Next, we need to find the final kinetic energy, which is zero because the runner's speed becomes zero by the time he reaches the base.
Therefore, the change in kinetic energy, ΔKE, is equal to:
ΔKE = 0 - KE_initial
Now we can calculate the mechanical energy lost due to friction using the formula:
Work = ΔKE = -μ * m * g * d
where μ is the coefficient of friction, m is the mass of the runner, g is the acceleration due to gravity (approximately 9.8 m/s^2), and d is the distance over which the friction acts.
We don't know the distance d yet, but we can solve for it using the work-energy principle. Since the work done by friction is equal to the mechanical energy lost, we can rewrite the equation as:
-μ * m * g * d = -ΔKE
Simplifying the equation:
d = (-ΔKE) / (μ * m * g)
Now we have all the information needed to calculate the answer to question (a).
To answer question (b), we need to calculate the distance over which the friction acts. We can use the formula derived above:
d = (-ΔKE) / (μ * m * g)
Let's calculate the answers now.
First, we need to calculate ΔKE:
ΔKE = 0 - KE_initial
Next, we calculate KE_initial:
KE_initial = 0.5 * mass * velocity^2
Then, we calculate the mechanical energy lost due to friction:
Work = ΔKE = -μ * m * g * d
Finally, we calculate the distance over which friction acts:
d = (-ΔKE) / (μ * m * g)
By substituting the given values and performing the calculations, we can find the answers to both parts of the question.