A 71.9-kg skier coasts up a snow-covered hill that makes an angle of 27.5 ° with the horizontal. The initial speed of the skier is 9.67 m/s. After coasting a distance of 1.50 m up the slope, the speed of the skier is 4.52 m/s. (a) Find the work done by the kinetic frictional force that acts on the skis. (b) What is the magnitude of the kinetic frictional force?

To determine the work done by the kinetic frictional force and the magnitude of the kinetic frictional force, we can follow these steps:

(a) To find the work done by the kinetic frictional force, we'll use the work-energy theorem. The work done by the force is equal to the change in kinetic energy.

The initial kinetic energy (K1) of the skier is given as 1/2 * m * v1^2, where m is the mass of the skier and v1 is the initial speed. Substituting the given values:

K1 = 1/2 * 71.9 kg * (9.67 m/s)^2

Next, we need to find the final kinetic energy (K2) of the skier. The final speed (v2) is given as 4.52 m/s. Using the same formula:

K2 = 1/2 * 71.9 kg * (4.52 m/s)^2

The change in kinetic energy ΔK = K2 - K1.

Finally, the work done by the kinetic frictional force (W) is equal to the change in kinetic energy:

W = ΔK = K2 - K1

(b) To find the magnitude of the kinetic frictional force (Fk), we need to apply Newton's second law of motion:

Fk = μ * N

The normal force (N) is equal to the component of the weight that acts perpendicular to the slope, which can be calculated as:

N = m * g * cos(θ)

where m is the mass of the skier, g is the acceleration due to gravity, and θ is the angle of the slope.

Once we have the normal force, we can calculate the kinetic frictional force (Fk) by multiplying it by the coefficient of kinetic friction (μ).

Now let's calculate the values step by step:

Step 1: Calculate the work done by the kinetic frictional force (a)

K1 = 1/2 * 71.9 kg * (9.67 m/s)^2
K2 = 1/2 * 71.9 kg * (4.52 m/s)^2
ΔK = K2 - K1

Step 2: Calculate the magnitude of the kinetic frictional force (b)

N = m * g * cos(θ)
Fk = μ * N

Substitute the known values and calculate the final results.