A 66.0 kg base runner begins his slide into second base when moving at a speed of 4.4 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?
J
(b) How far does he slide?
m
friction work done = ke lost = (1/2)m v^2
force = .7 m g = .7 * 66 * 9.8
mass = 66
a = F/m = .7 * 9.8 = 6.86 m/s^2
0 = 4.4 - 6.86 t
t = .641 seconds
average speed = 2.2 m/s
distance = 2.2 * .641 = 1.41 meters
To answer these questions, we need to use the concept of mechanical energy and the work-energy principle.
(a) We can find the mechanical energy lost due to friction by finding the work done by friction. The equation for work is given by:
Work = Force * Distance * cos(theta)
In this case, the force of friction is equal to the coefficient of friction (μ) multiplied by the normal force (which is the weight of the runner). The distance is the distance over which friction acts, and theta is the angle between the direction of the force and the direction of motion. In this case, since the runner is sliding horizontally, the angle theta is 0° and the cosine of 0° is 1.
So, the equation for work due to friction becomes:
Work = μ * m * g * d
where m is the mass of the runner, g is the acceleration due to gravity (9.8 m/s^2), and d is the distance over which friction acts.
To find the distance, we can use the equation of motion that relates initial velocity, final velocity, acceleration, and distance:
v^2 = u^2 + 2 * a * s
where v is the final velocity (0 m/s), u is the initial velocity (4.4 m/s), a is the acceleration (which is equal to the coefficient of friction multiplied by g), and s is the distance.
Rearranging for distance, we have:
s = (v^2 - u^2) / (2 * a)
Now we can substitute the given values into the equations to find the answers.
First, let's find the acceleration (a):
a = μ * g
Then we can find the distance (s). Plugging in the values:
s = (0^2 - 4.4^2) / (2 * 0.70 * 9.8)
Finally, we can calculate the work done by friction by multiplying the distance by the force of friction:
Work = μ * m * g * d
Substituting the values:
Work = 0.70 * 66.0 * 9.8 * s
(b) To find the distance, we can substitute the already calculated value of s into the equation:
Distance = s
By calculating these equations, we can find the answers to the questions.