A skier starts from rest at the top of a hill that is inclined at 12.0° with the horizontal. The hillside is 180.0 m long, and the coefficient of friction between the snow and the 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 move along the horizontal portion of the snow before coming to rest?
I kniw the answer is 323m but I don't understand how to get there!
To find the distance the skier moves along the horizontal portion of the snow before coming to rest, we need to analyze the forces acting on the skier.
1. Determine the gravitational force acting on the skier:
The skier's weight can be found by multiplying the mass by the acceleration due to gravity. However, we are not given the mass of the skier. Luckily, we can still work with the force of gravity using the equations:
Weight = mass x gravitational acceleration
Weight = m x g
The gravitational force acting on the skier can be split into two components: one parallel to the incline (mg sinθ) and one perpendicular to the incline (mg cosθ).
2. Calculate the force of friction acting on the skier:
The force of friction opposes the skier's motion and is equal to the coefficient of friction multiplied by the normal force (the force perpendicular to the incline). The normal force is equal to the perpendicular component of the weight force.
Frictional force = coefficient of friction x normal force
3. Determine the net force acting on the skier:
The net force is the vector sum of all the forces acting on the skier. In this case, the net force is the difference between the force parallel to the incline and the force of friction.
Net force = Force parallel to incline - Frictional force
4. Calculate the acceleration of the skier:
The net force can be used to determine the acceleration of the skier using Newton's second law (F = ma). In our case, the force parallel to the incline is acting as the net force:
Net force = m x a
Force parallel to incline - Frictional force = m x a
5. Calculate the time taken for the skier to reach the bottom of the hill:
Using one of the kinematic equations (s = ut + 0.5at^2), where s is the distance, u is the initial speed (which is 0 since the skier starts from rest), a is the acceleration, and t is the time taken.
In this case, the distance is the length of the hillside (180.0 m) and the acceleration is the one calculated using the net force.
6. Calculate the distance traveled on the horizontal portion of the snow:
The skier's velocity at the bottom of the hill is given by s = ut + 0.5at^2. The time spent on the horizontal portion of the snow can be determined by dividing the horizontal distance by the horizontal velocity.
Finally, multiplying the horizontal velocity by the time spent will give the distance traveled on the horizontal portion.
By following these steps, you should be able to find that the skier moves 323 m along the horizontal portion of the snow before coming to rest.
The friction force during the motion along the inclined plane is
F(fr)1 = kN = k•m•g•cosα.
The work of this force is
W1= k•m•g•cosα•s.
The law of conservation of energy gives
PE=KE+W1,
m•g•h = m•v^2/2 + k•m•g•cosα•s.
m•v^2/2 = m•g•h - k•m•g•cosα•s.
For the motion along the horizontal path
KE=W2,
m•v^2/2 = k•m•g•s1
s1= m•v^2/2• k•m•g