A 70.0 kg base runner begins his slide into second base while moving at a speed of 4.0 m/s. The coefficient of friction between his clothes and Earth is .70. He slides so that his speed is zero just as he reaches the base. How much mechanical energy is lost due to friction acting on the runner.
I know the equation I have to use is KE_i - PE_f = delta ME, but I don't know how to use the coefficient of friction in those equations.
All of the KE is lost due to friction. He had ke, now he has zero ke. Conclusion: all of the KE is dissipated in friction. The coefficent of friction is not needed for this part.
so the mechanical energy lost is just how much kinetic energy there was in the beginning?
I also need to find how far he slides, and the only equation that doesn't use time is vf^2=(vi)^2+2a*x, and I don't know acceleration either. What equation works?
Ok, you know the average force*distance=KE lost.
but force= mg*mu
solve for distance.
a. -560 J
b. d = 1.2 m
To find the mechanical energy lost due to friction, you can use the equation KE_i - PE_f = ΔME, as you mentioned. However, in this case, you don't need to use the coefficient of friction directly in this equation.
The initial kinetic energy, KE_i, can be calculated using the formula KE = 0.5 * mass * velocity^2. Plugging in the given values, we can calculate the initial kinetic energy:
KE_i = 0.5 * 70.0 kg * (4.0 m/s)^2
KE_i = 560 J
Since the runner comes to rest at the end of the slide, his final potential energy, PE_f, will be zero. Therefore, ΔME = KE_i - PE_f = 560 J.
The mechanical energy lost due to friction is equal to the initial kinetic energy, so the mechanical energy lost in this scenario is 560 J.
To find the distance the runner slides, you can use the equation vf^2 = vi^2 + 2a * x. Since the runner comes to rest, the final velocity, vf, is zero. The initial velocity, vi, is given as 4.0 m/s. Solving for x, we get:
0^2 = (4.0 m/s)^2 + 2a * x
Simplifying further, we can rearrange the equation to:
-16 m^2/s^2 = 2a * x
The acceleration, a, can be calculated using the formula a = μ * g, where μ is the coefficient of friction and g is the acceleration due to gravity (approximately 9.8 m/s^2). Plugging in the given coefficient of friction (.70), we find:
a = 0.70 * 9.8 m/s^2
a = 6.86 m/s^2
Now substituting the values back into the equation:
-16 m^2/s^2 = 2 * 6.86 m/s^2 * x
Simplifying further:
-16 m^2/s^2 = 13.72 m/s^2 * x
Dividing both sides by 13.72 m/s^2:
x = -1.16 m
Since distance cannot be negative in this context, we take the absolute value and conclude that the runner slides approximately 1.16 meters before coming to rest.
To find the mechanical energy lost due to friction, you are correct that you need to use the equation KE_i - PE_f = ΔME. However, in this case, the potential energy (PE) can be assumed to be constant, since there is no change in height. Therefore, the equation simplifies to KE_i = ΔME.
Since the runner slides to a stop, all of the initial kinetic energy (KE_i) is lost due to friction. The coefficient of friction is not needed to calculate the mechanical energy lost. So, to find the mechanical energy lost, you can simply use the initial kinetic energy of the runner.
Now, to find how far the runner slides, you can use the equation vf^2 = vi^2 + 2a∆x, where vf is the final velocity (which is zero in this case), vi is the initial velocity (given as 4.0 m/s), a is the acceleration, and ∆x is the distance the runner slides.
Since the runner comes to a stop, the final velocity (vf) is zero, and plugging the known values into the equation gives you:
0 = (4.0 m/s)^2 + 2a∆x
Simplifying the equation:
0 = 16.0 m^2/s^2 + 2a∆x
From this equation, you can solve for ∆x, which is the distance the runner slides.
To find the acceleration (a), you can use the equation force = mass * acceleration. In this case, the force is the force of friction, which can be calculated as the product of the normal force (mg) and the coefficient of friction (μ). So, you have:
force = (70.0 kg) * (9.8 m/s^2) * 0.70
Now, solve this equation for acceleration (a). Once you have the value of a, plug it back into the previous equation to solve for ∆x, the distance the runner slides.
I hope this explanation helps you understand how to solve the problem.