A skier with a mass of 66.6 kg starts from rest and skis down an icy (frictionless) slope that has a length of 58.4 m at an angle of 29.8° 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. What is the coefficient of kinetic friction between the skier and the horizontal surface?
To solve this problem, we need to consider the different stages of the skier's motion and use the principles of physics.
First, let's find the initial potential energy of the skier at the top of the slope.
Potential Energy = mass * gravity * height
Potential Energy = (66.6 kg) * (9.8 m/s^2) * (58.4 m * sin(29.8°))
Next, let's find the final velocity of the skier at the bottom of the slope using the principle of conservation of mechanical energy.
Initial Potential Energy = Final Kinetic Energy
0.5 * mass * velocity^2 = (66.6 kg) * (9.8 m/s^2) * (distance down the slope)
velocity = sqrt((2 * (66.6 kg) * (9.8 m/s^2) * (58.4 m * sin(29.8°)))/(66.6 kg))
Next, let's use this final velocity to find the time it takes for the skier to come to rest on the horizontal path.
Time = (distance on horizontal path) / (final velocity)
Time = 160 m / (sqrt((2 * (66.6 kg) * (9.8 m/s^2) * (58.4 m * sin(29.8°)))/(66.6 kg)))
Finally, let's use this time to find the coefficient of kinetic friction between the skier and the horizontal surface using the equation of motion.
distance = initial velocity * time - 0.5 * coefficient of kinetic friction * gravity * time^2
0 = (final velocity) * t - 0.5 * (coefficient of kinetic friction) * (9.8 m/s^2) * t^2
Simplify and solve for the coefficient of kinetic friction:
coefficient of kinetic friction = (final velocity) / (4.9 m/s^2 * t)
By plugging in the values calculated for the final velocity and time, we can find the coefficient of kinetic friction between the skier and the horizontal surface.
To find the coefficient of kinetic friction between the skier and the horizontal surface, we need to analyze the forces acting on the skier.
1. Determine the acceleration down the slope:
The skier experiences an acceleration due to the gravitational force acting along the incline. The component of the gravitational force parallel to the incline is:
F_parallel = m * g * sin(θ)
where m is the mass of the skier (66.6 kg), g is the acceleration due to gravity (9.8 m/s²), and θ is the angle of the slope (29.8°).
So, the acceleration down the slope (a) is given by:
a = F_parallel / m
2. Find the velocity at the bottom of the slope:
We can use the kinematic equation relating distance, initial velocity (which is zero), and acceleration:
v^2 = u^2 + 2as
where v is the final velocity at the bottom of the slope, u is the initial velocity (zero in this case), a is the acceleration down the slope, and s is the length of the slope (58.4 m).
Rearranging the equation, we get:
v = sqrt(2 * a * s)
3. Determine the deceleration on the horizontal path:
Once the skier reaches the horizontal path, the forces on the skier change. The deceleration on the horizontal path is caused by the force of kinetic friction.
The force of kinetic friction is given by:
F_kinetic = m * a_kinetic
where m is the mass of the skier, and a_kinetic is the deceleration on the horizontal path.
Since the skier comes to rest, the final velocity is zero.
We can use the kinematic equation again to relate distance, final velocity (zero), initial velocity (v from step 2), and acceleration:
v^2 = u^2 + 2as
where u is the initial velocity (v from step 2), a is the deceleration on the horizontal path, and s is the distance the skier travels on the horizontal path (160 m).
Rearranging the equation, we get:
a = -v^2 / (2 * s)
(Note the negative sign because the acceleration is opposite to the initial velocity.)
4. Calculate the kinetic friction coefficient:
The force of kinetic friction can be determined as:
F_kinetic = μ_kinetic * m * g
where μ_kinetic is the coefficient of kinetic friction, m is the mass of the skier, and g is the acceleration due to gravity.
Equating the forces of kinetic friction from step 3 and step 4, we have:
m * a_kinetic = μ_kinetic * m * g
Cancelling out the mass m, we get:
a_kinetic = μ_kinetic * g
Rearranging the equation, we find:
μ_kinetic = a_kinetic / g
Now, we have all the necessary information to find the coefficient of kinetic friction. Let's substitute the values into the equations and calculate the coefficient.