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.
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.
A. To find the coefficient of kinetic friction between the skis and the snow, we need to use the work-energy principle:
The work-energy principle states that the work done on an object is equal to the change in its kinetic energy.
The work done by the force of friction can be calculated using the formula:
Work = Force x Distance x cos(θ)
Where:
Force = frictional force = coefficient of kinetic friction (μ) x normal force (N)
Distance = distance traveled by the skier (169 m)
θ = angle between the force of friction and the direction of motion (θ = 180 degrees for an object coming to a stop on level ground)
Normal force (N) = weight of the skier = mass (m) x acceleration due to gravity (g)
The change in kinetic energy can be calculated using the formula:
ΔKE = KE_final - KE_initial
= 0 - 1/2 x mass x velocity^2
Since the skier comes to a stop, the final kinetic energy is 0.
Therefore, we have:
Work = ΔKE
Now, let's substitute the values and solve for the coefficient of kinetic friction (μ):
Work = Force x Distance x cos(θ)
= μN x Distance x cos(θ)
ΔKE = 0 - 1/2 x mass x velocity^2
= -1/2 x mass x velocity^2
Since the normal force (N) is equal to the weight of the skier:
N = mass x acceleration due to gravity
= m x g
Substituting the value of N into the equation:
Work = μN x Distance x cos(θ)
= μ x m x g x Distance x cos(θ)
ΔKE = -1/2 x mass x velocity^2
Equating work and change in kinetic energy:
μ x m x g x Distance x cos(θ) = -1/2 x mass x velocity^2
Simplifying the equation:
μ x g x Distance x cos(θ) = -1/2 x velocity^2
Now let's substitute the known values and solve for μ:
μ x 9.8 m/s^2 x 169 m x cos(180 degrees) = -1/2 x (2.4 m/s)^2
μ x 9.8 m/s^2 x 169 m x (-1) = -1/2 x (2.4 m/s)^2
μ x 9.8 m/s^2 x 169 m = 1/2 x (2.4 m/s)^2
Finally, solving for μ:
μ = (1/2 x (2.4 m/s)^2) / (9.8 m/s^2 x 169 m)
μ ≈ 0.064
Therefore, the coefficient of kinetic friction between the skis and the snow is approximately 0.064.
B. If a 70.0 kg skier with twice the starting speed coasted the same distance before stopping, we can use the same equation to find the coefficient of kinetic friction (μ).
In this case, the mass (m) is 70.0 kg and the velocity (v) is 2 x 2.4 m/s = 4.8 m/s.
We can use the equation:
μ x g x Distance x cos(θ) = -1/2 x velocity^2
to find μ.
Substituting the known values:
μ x 9.8 m/s^2 x 169 m x cos(180 degrees) = -1/2 x (4.8 m/s)^2
μ x 9.8 m/s^2 x 169 m x (-1) = -1/2 x (4.8 m/s)^2
μ x 9.8 m/s^2 x 169 m = 1/2 x (4.8 m/s)^2
Simplifying the equation:
μ = (1/2 x (4.8 m/s)^2) / (9.8 m/s^2 x 169 m)
μ ≈ 0.032
Therefore, the coefficient of kinetic friction between the skis and the snow for the 70.0 kg skier with twice the starting speed is approximately 0.032.
To solve parts A and B, we need to use the work-energy principle, which states that the work done on an object is equal to its change in kinetic energy.
A. Let's start with part A.
Step 1: Find the change in kinetic energy
The initial kinetic energy of the skier is given by 1/2 * mass * (initial velocity)^2.
The final kinetic energy of the skier is zero since the skier comes to a stop.
Initial kinetic energy = 1/2 * 60.0 kg * (2.40 m/s)^2 = 172.8 J
Final kinetic energy = 0 J
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
= 0 J - 172.8 J
= -172.8 J
Step 2: Find the work done by friction
The work done by the friction force can be calculated as force of friction * distance.
The force of friction is the product of the coefficient of kinetic friction (μk) and the normal force (which is equal to the weight of the skier, mg).
Work done by friction = force of friction * distance
We know that work done by friction = Change in kinetic energy, so our equation becomes:
μk * mg * distance = -172.8 J
Step 3: Solve for the coefficient of kinetic friction (μk)
μk = (-172.8 J)/(mg * distance)
Substituting the given values into the equation:
μk = (-172.8 J)/(60.0 kg * 9.8 m/s^2 * 169 m)
Now calculate μk.
B. For part B, we need to find the coefficient of kinetic friction when a 70.0 kg skier with twice the starting speed coasts the same distance before stopping.
Step 1: Find the change in kinetic energy
In this case, the initial velocity is 2 * 2.40 m/s = 4.80 m/s.
Initial kinetic energy = 1/2 * 70.0 kg * (4.80 m/s)^2
Final kinetic energy = 0 J
Change in kinetic energy = Final kinetic energy - Initial kinetic energy
Step 2: Find the work done by friction
The work done by friction is still equal to the change in kinetic energy.
Step 3: Solve for the coefficient of kinetic friction (μk)
Using the same formula as before:
μk * mg * distance = change in kinetic energy
Substituting the given values, calculate μk.
That's it! By following these steps, you can find the coefficients of kinetic friction for both parts A and B.