A 60.0 kg skier on level snow coasts 169 m to a stop from a speed of 2.40 m/s.
A. Use the work-energy principle to find the coefficient of kinetic friction between the skis and the snow.
B. Suppose a 70.0 kg skier with twice the starting speed coasted the same distance before stopping. Find the coefficient of kinetic friction between that skier's skis and the snow.
To find the coefficient of kinetic friction between the skis and the snow, we can use the work-energy principle. According to this principle, the work done on an object is equal to the change in its kinetic energy.
A. Let's start by finding the work done on the skier. The work done by friction can be calculated using the equation:
Work = Force x Distance x cosine(θ)
Since the skier is moving on level snow, the angle θ between the force of friction and the displacement is 0 degrees, and the cosine of 0 is 1. So we can simplify the equation:
Work = Force x Distance
The work done is equal to the change in kinetic energy:
Work = ΔKE
The initial kinetic energy (KE_initial) can be calculated using the equation:
KE_initial = (1/2) x mass x (initial velocity)^2
KE_initial = (1/2) x 60.0 kg x (2.40 m/s)^2
Next, we need to determine the final kinetic energy (KE_final). Since the skier comes to a stop, the final kinetic energy is zero (KE_final = 0).
Therefore, the work done on the skier is:
Work = ΔKE = KE_final - KE_initial = 0 - (1/2) x 60.0 kg x (2.40 m/s)^2
Now, let's calculate the work done by friction. We can use the equation:
Work = Force x Distance
The distance is given as 169 m, and we need to find the force of friction.
Work = Force x 169 m
Now, equating the two expressions for work:
Force x 169 m = 0 - (1/2) x 60.0 kg x (2.40 m/s)^2
Simplifying:
Force = - [(1/2) x 60.0 kg x (2.40 m/s)^2] / 169 m
Finally, let's find the coefficient of kinetic friction (μ). The force of friction can be calculated using the equation:
Force = μ x Normal Force
Since the skier is on level ground, the normal force (N) is equal to the weight of the skier, which can be calculated as:
Weight = mass x gravity
Weight = 60.0 kg x 9.8 m/s^2
Now, equating the force of friction to the product of the coefficient of kinetic friction and the weight:
μ x (60.0 kg x 9.8 m/s^2) = - [(1/2) x 60.0 kg x (2.40 m/s)^2] / 169 m
Simplifying:
μ = - [(1/2) x 60.0 kg x (2.40 m/s)^2] / (169 m x 60.0 kg x 9.8 m/s^2)
Now, perform the calculations to find the coefficient of kinetic friction.
B. To find the coefficient of kinetic friction for a 70.0 kg skier with twice the starting speed, we can use the same formula as before. The only difference is the initial velocity, which is now 2.40 m/s x 2 = 4.80 m/s.
Repeat the calculations using the new values for mass and initial velocity to find the coefficient of kinetic friction for the second skier.