snowboarder slides down a 37° slope at a constant 4 m/s. What is the coefficient of kineticfriction?

1. Break Down the Problem

To find the coefficient of kinetic friction ($\mu_k$), we can use Newton's second law. We need to analyze the forces acting on the snowboarder. The key forces are:

1. The gravitational force acting downward.
2. The normal force acting perpendicular to the slope.
3. The frictional force acting opposite to the direction of motion.

2. Relevant Concepts

• The gravitational force can be resolved into two components:

  • Parallel to the slope: $F_{\text{gravity, parallel}} = mg \sin(\theta)$
  • Perpendicular to the slope: $F_{\text{gravity, perpendicular}} = mg \cos(\theta)$

• The normal force $N$ is equal to the perpendicular component of the gravitational force: $N = mg \cos(\theta)$

• The frictional force ($F_{f}$) can also be expressed in terms of the coefficient of kinetic friction: $F_{f} = \mu_k N$

• Since the snowboarder is sliding down at a constant speed, the net force along the slope is zero: $F_{\text{gravity, parallel}} - F_f = 0$ Thus: $mg \sin(\theta) - \mu_k mg \cos(\theta) = 0$

3. Analysis and Detail

Simplifying the equation gives: $mg \sin(\theta) = \mu_k mg \cos(\theta)$

Dividing both sides by $mg$ (assuming $m \neq 0$): $\sin(\theta) = \mu_k \cos(\theta)$

Solving for the coefficient of kinetic friction: $\mu_k = \frac{\sin(\theta)}{\cos(\theta)} = \tan(\theta)$

Given that $\theta = 37^\circ$: $\mu_k = \tan(37^\circ)$

Using the known value: $\tan(37^\circ) \approx 0.7536$

4. Verify and Summarize

We can verify that the calculation accurately reflects the relationships in the forces at play, taking into account the snowboarder’s constant speed. The coefficient of kinetic friction should be representative of the interaction between the snowboard and the slope surface.

Final Answer

The coefficient of kinetic friction is approximately: $\mu_k \approx 0.7536$