Sam, whose mass is 72.0 kg, takes off down a 51.0 m high, 11.0^\circ slope on his jet-powered skis. The skis have a thrust of 180 N. Sam's speed at the bottom is 43.0 {\rm m/s}. What is the coefficient of kinetic friction of his skis on snow?

To find the coefficient of kinetic friction between Sam's skis and the snow, we can use the concept of work and energy.

Step 1: Calculate the gravitational potential energy at the top of the slope.
The gravitational potential energy (GPE) can be calculated using the formula: GPE = mgh, where m is the mass (72.0 kg), g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height (51.0 m). Therefore, GPE = 72.0 kg × 9.8 m/s² × 51.0 m.

GPE = 35328 J

Step 2: Calculate the work done by the thrust force.
The work done (W) can be calculated using the formula: W = Fd, where F is the force (180 N) and d is the distance (51.0 m). Therefore, W = 180 N × 51.0 m.

W = 9180 J

Step 3: Calculate the work done against friction.
Since the net work is zero (no change in kinetic energy), the work done against friction is equal to the work done by the thrust force (W = 9180 J).

Step 4: Calculate the friction force.
The friction force (f) can be calculated using the formula: f = μN, where μ is the coefficient of kinetic friction and N is the normal force. The normal force is equal to the gravitational force acting perpendicular to the slope, which can be calculated as mgcos(θ), where θ is the angle of the slope (11.0°). Therefore, N = mgcos(θ) = 72.0 kg × 9.8 m/s² × cos(11.0°).

N ≈ 705.27 N

Substituting the values into the formula f = μN, we have 9180 J = μ × 705.27 N.

Step 5: Calculate the coefficient of kinetic friction.
Rearranging the equation, μ = (9180 J) / (705.27 N). Calculating this value gives us the coefficient of kinetic friction.

μ ≈ 13.02

Therefore, the coefficient of kinetic friction of Sam's skis on the snow is approximately 13.02.