Ski boots are usually made with releasable bindings. This is done for safety – if you fall and your ski gets caught, the binding between the boot and the ski will release so the ski pops off rather than twisting your knee. Sometimes this leads to the embarrassing situation of you falling down while skiing, the ski popping off, and the ski continuing down the mountain without you. You're skiing down a slope angled at 15∘ with respect to the horizontal when this situation happens to you. Your ski continues down the slope at a constant speed of 4 m/s. What is the coefficient of friction between your ski and the slope?

Details and assumptions
The acceleration of gravity is −9.8 m/s2.
Neglect air resistance.
You may assume this has never happened to your challenge master. Honest.

To find the coefficient of friction between the ski and the slope, we can use the concept of the forces acting on the ski. In this case, the only force acting on the ski is the force due to friction.

Let's break down the forces involved:

1. The force due to gravity: The weight of the ski pulling it downward is given by the formula W = mg, where m is the mass of the ski and g is the acceleration due to gravity.

2. The normal force: This is the force exerted by the slope on the ski in the vertical direction. It is equal in magnitude and opposite in direction to the weight of the ski.

3. The force due to friction: This is the force that opposes the motion of the ski down the slope. The coefficient of friction (μ) is the ratio of the force of friction to the normal force.

Since the ski is moving down the slope at a constant speed, we can assume that the force due to friction (f) is equal in magnitude but opposite in direction to the component of the weight of the ski acting parallel to the slope. Let's calculate the magnitude of this force:

The component of the weight acting down the slope is given by W_parallel = mg sin(θ), where θ is the angle of the slope.

Since the ski is moving at a constant speed, the force due to friction is equal in magnitude but opposite in direction, so we have:

f = mg sin(θ)

Now, we can calculate the normal force:

The normal force (N) is equal to the weight of the ski, so we have:

N = mg

Finally, we can substitute the values into the equation for the coefficient of friction:

μ = f / N = (mg sin(θ)) / (mg) = sin(θ)

Given that the slope angle θ is 15 degrees, we can calculate the coefficient of friction:

μ = sin(15∘)

Using a calculator, we find:

μ ≈ 0.259

Therefore, the coefficient of friction between your ski and the slope is approximately 0.259.