A 69-kg base runner begins his slide into second base when he is moving at a speed of 3.7 m/s. The coefficient of friction between his clothes and Earth is 0.70. 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?
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To find the answers to both (a) and (b), we need to apply the principles of work, energy, and friction. Let's break it down step by step.
(a) To calculate the amount of mechanical energy lost due to friction acting on the runner, we need to find the initial kinetic energy of the runner and the final kinetic energy at the point when he reaches the base.
The initial kinetic energy (KE1) can be calculated using the formula:
KE1 = (1/2) * mass * velocity^2
Given that the mass of the runner is 69 kg and his initial speed is 3.7 m/s, we can plug these values into the formula:
KE1 = (1/2) * 69 kg * (3.7 m/s)^2
Simplifying the equation:
KE1 = 0.5 * 69 kg * (3.7 m/s)^2
Now, let's find the final kinetic energy (KE2) when the runner's speed is zero. Since the speed is zero at this point, the final kinetic energy is also zero.
The mechanical energy lost due to friction can be calculated as the difference between the initial and final kinetic energies:
Energy lost due to friction = KE1 - KE2
Since KE2 is zero, the energy lost due to friction is equal to KE1.
Calculating the result:
Energy lost due to friction = 0.5 * 69 kg * (3.7 m/s)^2
(b) To find how far the runner slides, we can use the work-energy principle and the concept of frictional work.
The work done by friction (Wf) can be calculated using the formula:
Wf = frictional force * distance
The frictional force (F) is given by the product of the coefficient of friction (μ) and the normal force (N). In this case, the normal force is equal to the runner's weight, which can be found using the formula:
N = mass * gravity
Given that the coefficient of friction is 0.70 and the mass is 69 kg, the normal force (N) can be calculated as:
N = 69 kg * 9.8 m/s^2
Now we can calculate the frictional force (F):
F = coefficient of friction * normal force
After finding F, we can use the work-energy principle, which states that the work done by friction is equal to the change in mechanical energy:
Work done by friction = Energy lost due to friction
Now, we can rearrange the equation to solve for the distance:
Distance = Work done by friction / frictional force
Calculating the result:
Distance = (0.5 * 69 kg * (3.7 m/s)^2) / (0.70 * (69 kg * 9.8 m/s^2))
By substituting the given values into the formula, you will get the answer for both (a) and (b).