A skier with a mass of 70 kg starts from rest and skis down an icy (frictionless) slope that has a length of 52 m at an angle of 32° with respect to the horizontal. At the bottom of the slope, the path levels out and becomes horizontal, the snow becomes less icy, and the skier begins to slow down, coming to rest in a distance of 160 m along the horizontal path.
(a) What is the speed of the skier at the bottom of the slope?
m/s
(b) What is the coefficient of kinetic friction between the skier and the horizontal surface?
µk =
I will be happy to critique your thinking.
I tried using free fall equations with no luck. I guess this isnt a free fall problem?
That reply was meant for a different question.
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To find the speed of the skier at the bottom of the slope, we can use the principle of conservation of energy. The initial potential energy of the skier at the top of the slope is converted into kinetic energy as they slide down the slope.
(a) Calculate the potential energy at the top of the slope:
Potential Energy = mass × acceleration due to gravity × height
= 70 kg × 9.8 m/s² × (52 m × sin(32°))
Next, calculate the kinetic energy at the bottom of the slope. Since the path levels out and becomes horizontal, there is no change in height, so the potential energy is zero. Thus, the entire potential energy is converted into kinetic energy.
Kinetic Energy = Potential Energy at the top of the slope
Therefore, we can calculate the speed using the formula for kinetic energy:
Kinetic Energy = 0.5 × mass × velocity²
Substituting the values, we have:
0.5 × mass × velocity² = Potential Energy at the top of the slope
Solving for velocity:
velocity = sqrt((2 × Potential Energy at the top of the slope) / mass)
(b) To find the coefficient of kinetic friction between the skier and the horizontal surface, we can use the equation of motion. The skier comes to rest in a distance of 160 m along the horizontal path. Using the equation:
(v² - u²) = 2 × acceleration × distance
Here, 'v' is the final velocity (0 in this case), 'u' is the initial velocity (velocity at the bottom of the slope), and 'distance' is the given distance of 160 m.
Simplifying the equation, we have:
(0 - velocity²) = 2 × acceleration × 160 m
Rearranging and solving for acceleration:
acceleration = (-velocity²) / (2 × distance)
The force of friction acting on the skier on the horizontal surface is given by:
Force of friction = mass × acceleration
Finally, we can calculate the coefficient of kinetic friction using the equation:
Force of friction = coefficient of kinetic friction × mass × acceleration
Substituting the values, we can solve for the coefficient of kinetic friction.