A speed skier is travelling horizontally at a constant speed of 5.50 m/s when she approaches a snow-covered hill that has a slope of 24.0° above the horizontal. The coefficient of kinetic friction between the skis and the hill is 0.110 and the combined mass of her and her skis is 80.0 kg. If she decided to glide up the hill, how far would she make it before she comes to a complete stop?

Question

A speed skier is travelling horizontally at a constant speed of 5.50 m/s when she approaches a snow-covered hill that has a slope of 24.0° above the horizontal. The coefficient of kinetic friction between the skis and the hill is 0.110 and the combined mass of her and her skis is 80.0 kg. If she decided to glide up the hill, how far would she make it before she comes to a complete stop?

Answer 1

To find how far the speed skier will make it before coming to a complete stop, we will first calculate the net force acting on the skier and then use that to determine the distance traveled. Here's how we can do it.

Step 1: Calculate the gravitational force acting on the skier:
The gravitational force can be calculated using the formula:
Force (gravity) = mass x gravitational acceleration
Force (gravity) = 80.0 kg x 9.8 m/s^2
Force (gravity) = 784 N

Step 2: Calculate the frictional force acting on the skier:
The frictional force can be calculated using the formula:
Force (friction) = coefficient of friction x normal force
The normal force can be calculated using the formula:
Normal force = mass x gravitational acceleration x cos(angle)
Normal force = 80.0 kg x 9.8 m/s^2 x cos(24.0°)
Normal force ≈ 703.77 N

Now, calculate the frictional force:
Force (friction) = 0.110 x 703.77 N
Force (friction) ≈ 77.42 N

Step 3: Calculate the net force:
The net force acting on the skier can be calculated using the formula:
Net force = Force (gravity) - Force (friction)
Net force = 784 N - 77.42 N
Net force = 706.58 N

Step 4: Calculate the distance traveled:
The work done by the net force is converted into kinetic energy, given by the formula:
Work = Kinetic energy = 1/2 x mass x velocity^2
We can rearrange this equation to find distance traveled:
Distance = (Force x distance) / (Force x acceleration)
Distance = (706.58 N x distance) / (706.58 N x 5.50 m/s)
Distance = (distance) / 5.50

Now, let's solve for distance:
Distance = (706.58 N x distance) / (706.58 N x 5.50 m/s)
1 = distance / 5.50 m/s
distance = 5.50 m/s

Therefore, the speed skier would make it approximately 5.50 meters up the hill before coming to a complete stop.