A 66.0 kg base runner begins his slide into second base when he is moving at a speed of 4 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?
Wouldn't it be equal the the KEnergy lost? 1/2 mv^2. Then to get the distance, KE=mu*mg*distance.
Thanks for your quick help
bobpursley, what does the mu stand for in your equation?
Mu is the coefficient of friction (μ).
556
To answer these questions, we need to consider the concept of work and energy. We can use the work-energy theorem, which states that the change in kinetic energy of an object is equal to the net work done on it.
(a) The first question asks how much mechanical energy is lost due to friction acting on the runner. We can calculate this by finding the change in kinetic energy of the runner.
The initial kinetic energy of the runner is given by the formula:
KE_initial = (1/2) * mass * velocity^2
where:
mass = 66.0 kg (mass of the runner)
velocity = 4 m/s (initial speed of the runner)
Plugging in the values, we can calculate the initial kinetic energy:
KE_initial = (1/2) * 66.0 kg * (4 m/s)^2
Next, we need to find the final kinetic energy. Given that the speed of the runner is zero when he reaches the base, the final kinetic energy is zero:
KE_final = 0
The change in kinetic energy is then:
ΔKE = KE_final - KE_initial
Substituting the values, we can calculate the change in kinetic energy:
ΔKE = 0 - [(1/2) * 66.0 kg * (4 m/s)^2]
This will give us the amount of mechanical energy lost due to friction acting on the runner.
(b) The second question asks how far the runner slides. We can use the work-energy theorem again to calculate the work done by the friction force.
The work done by the friction force is equal to the change in kinetic energy (due to the work-energy theorem):
Work = ΔKE
Now, let's consider the equation for work done by friction:
Work = force * distance * cos(θ)
where:
force = frictional force
distance = distance traveled by the runner (what we're trying to find)
θ = angle between the force and the displacement (which we can assume to be zero, since the force and displacement are in the same direction)
If we rearrange the equation, we get:
distance = Work / force
We already know the work done (which is ΔKE) and the force can be calculated using the equation:
force = coefficient of friction * normal force
The normal force is equal to the weight of the runner, which can be calculated using:
weight = mass * gravitational acceleration
where:
mass = 66.0 kg (mass of the runner)
gravitational acceleration = 9.8 m/s^2
Using the calculated force and the known work done, we can now find the distance traveled by the runner.
To summarize:
(a) The mechanical energy lost due to friction acting on the runner can be calculated by finding the change in kinetic energy of the runner: ΔKE = KE_final - KE_initial.
(b) The distance the runner slides can be calculated using the equation distance = Work / force, where the work done is equal to the change in kinetic energy (ΔKE) and the force can be calculated using the equation force = coefficient of friction * normal force.