A skier starts from rest at the top of a hill that is inclined 18.5° with respect to the horizontal. The hillside is 200 m long, and the coefficient of friction between snow and skis is μk = 0.0500. 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?
The force of friction along the slope is mu*mg*CosTheta, so energy lost is that times 200m
The starting energy is at the top of the hill, mass*g*200/sin18.5
So, the KE at the bottom of the hill is the difference between those two, and the final energy is in friction or
equal to mu*mg*Distance, and you can then solve for Distance.
how did you find mass
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.
Let's break down the problem into different stages:
Stage 1: Skier on the Inclined Hill
1. Calculate the gravitational force acting on the skier:
- The force of gravity is given by Fg = m * g, where m is the mass of the skier and g is the acceleration due to gravity (approximately 9.8 m/s²).
2. Calculate the normal force perpendicular to the hill:
- The normal force, Fn, is equal to the component of the gravitational force perpendicular to the hill, which is Fn = m * g * cos(θ), where θ is the angle of inclination (18.5° in this case).
3. Calculate the frictional force opposing the skier's motion:
- The frictional force, Ff, is given by Ff = μk * Fn, where μk is the coefficient of friction between the snow and skis.
4. Calculate the net force acting on the skier:
- The net force, Fnet, is equal to Fnet = Fg * sin(θ) - Ff. The first term represents the component of the gravitational force parallel to the hill, and the second term is the frictional force opposing the motion.
5. Calculate the acceleration of the skier:
- The acceleration, a, is given by a = Fnet / m, where m is the mass of the skier.
6. Calculate the time it takes for the skier to reach the bottom of the hill:
- The time, t, is determined using the kinematic equation s = ut + 0.5at², where u is the initial velocity of the skier along the hill (which is assumed to be 0 since the skier starts from rest) and s is the distance traveled along the hill (200 m in this case).
Stage 2: Skier on the Level Snow
After reaching the bottom of the hill, the skier glides along the horizontal portion of the snow.
7. Calculate the frictional force opposing the skier's motion:
- The frictional force, Ff_hor, is given by Ff_hor = μk * Fn_hor, where μk is the coefficient of friction between the snow and skis, and Fn_hor is the normal force acting on the skier now that the hill is level.
8. Calculate the acceleration of the skier:
- The acceleration, a_hor, can be determined using the equation a_hor = Ff_hor / m, where m is the mass of the skier.
9. Calculate the distance the skier glides along the horizontal portion:
- Since the skier comes to rest, the final velocity, v_hor, is 0. We can use the equation v² = u² + 2as_hor, where u is the initial velocity (which is the same as the skier's velocity at the bottom of the hill) and s_hor is the distance traveled along the horizontal portion.
By following these steps and plugging in the given values, you can find the distance the skier glides along the horizontal portion of the snow before coming to rest.