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 did he slide?
To answer the given question, we need to use the concept of work and energy, as well as the equations of motion. Let's break it down into two parts.
(a) To find out how much mechanical energy is lost due to friction acting on the runner, we can use the work-energy principle. The work done by friction is equal to the change in mechanical energy.
The initial mechanical energy of the runner is given by the formula:
E_initial = 1/2 * m * v_initial^2
where m is the mass of the runner (85.0 kg), and v_initial is the initial velocity (2.10 m/s).
The final mechanical energy of the runner is zero when his speed is zero at the base.
E_final = 0
The work done by friction (W_friction) is equal to the change in mechanical energy:
W_friction = E_final - E_initial
Now, let's calculate it:
E_initial = 1/2 * 85.0 kg * (2.10 m/s)^2
W_friction = 0 - E_initial
Therefore, the amount of mechanical energy lost due to friction acting on the runner is equal to W_friction.
(b) To determine how far the runner slid, we can use the equations of motion. The initial speed of the runner is given, and we need to find the distance covered before coming to a stop.
We can use the equation:
v_final^2 = v_initial^2 + 2 * a * d
where v_final is the final velocity (which is zero when the runner stops), a is the acceleration, and d is the distance covered by the runner.
Since the runner slides until he comes to a stop, his final speed is zero (v_final = 0), and the equation becomes:
0 = (2.10 m/s)^2 + 2 * a * d
We can rearrange this equation to solve for the distance d:
d = -(v_initial^2) / (2 * a)
To find the acceleration, we can use the equation of friction:
frictional force = coefficient of friction * normal force
The normal force is equal to the weight of the runner:
normal force = m * g
where g is the acceleration due to gravity (approximately 9.81 m/s^2).
The frictional force is equal to the mass of the runner multiplied by the acceleration:
frictional force = m * a
Equating the two expressions for the frictional force, we have:
m * a = coefficient of friction * m * g
Simplifying, we find:
a = coefficient of friction * g
Now we can substitute the values to calculate the distance d:
d = - (2.10 m/s)^2 / (2 * coefficient of friction * g)
Where:
m = 85.0 kg (mass of the runner)
v_initial = 2.10 m/s (initial speed of the runner)
coefficient of friction = 0.570 (friction between his clothes and Earth)
g = 9.81 m/s^2 (acceleration due to gravity)
Calculating these values will give us the distance covered by the runner.