A skier starts from rest at the top of a hill that is inclined at 10.3° with the horizontal. The hillside is 210 m long, and the coefficient of friction between snow and skis is 0.0750. At the bottom of the hill, the snow is level and the coefficient of friction is unchanged. How far does the skier glide along the horizontal portion of the snow before coming to rest?

energy from gravity: mgh=mg(210sin10.3)

energy into snow: 210*mg*cos10.3+mgL
set the two energies equal, solve for L.

oops, I forget mu coefficent friction.

energy in snow= 210*mu*mg*cos10.3 +mu*mgL

To find out how far the skier glides along the horizontal portion of the snow before coming to rest, we need to consider the forces acting on the skier.

Here are the steps to solve the problem:

Step 1: Calculate the gravitational force component parallel to the hill.
Since the hill is inclined at an angle of 10.3° with the horizontal, the component of the gravitational force acting parallel to the hill is given by:
Force_parallel = m * g * sin(theta)
where m is the mass of the skier, g is the acceleration due to gravity (approximately 9.8 m/s²), and theta is the angle of the hill (10.3°).

Step 2: Calculate the frictional force acting on the skier while on the hill.
The frictional force can be calculated using the equation:
Force_friction = m * g * cos(theta) * mu
where mu is the coefficient of friction between the skis and the snow.

Step 3: Find the net force acting on the skier while on the hill.
The net force can be found by subtracting the frictional force from the gravitational force component parallel to the hill.
Net_force = Force_parallel - Force_friction

Step 4: Calculate the acceleration of the skier on the hill.
The acceleration can be found using Newton's second law of motion:
Acceleration = Net_force / m

Step 5: Calculate the time it takes for the skier to reach the bottom of the hill.
We can use the equation of motion:
Distance = (1/2) * acceleration * time²
where Distance is the length of the hill (210 m in this case).

Step 6: Calculate the final velocity of the skier at the bottom of the hill.
The final velocity can be calculated using the equation:
Final_velocity = initial_velocity + acceleration * time

Step 7: Calculate the distance the skier glides along the horizontal snow before coming to rest.
The skier will come to rest when the net force acting on them is zero. So, the distance they glide along the horizontal portion of the snow can be calculated using:
Distance_horizontal = (Final_velocity)² / (2 * frictional force on level snow)

By following these steps, you can determine how far the skier glides along the horizontal portion of the snow before coming to rest.