A skier skiing downhill reaches the bottom of a hollow with a velocity of 20.0m/s, and then coasts up a hill with a 12 degree slope. If the coefficient of kinetic friction is 0.11, how far up the slope will she travel before she stops?
To find how far up the slope the skier will travel before stopping, we need to analyze the forces acting on the skier.
First, let's consider the downhill part of the skier's motion. As the skier reaches the bottom of the hollow, the only force acting on the skier is the force of kinetic friction opposing the motion. We can calculate this force using the equation:
frictional force = coefficient of kinetic friction * normal force
The normal force is the force exerted by the surface perpendicular to it, which is equal to the skier's weight, given by:
weight = mass * gravity
Now, let's calculate the frictional force. Assuming the skier's mass is m, the frictional force can be written as:
frictional force = 0.11 * (m * gravity)
The work done by the frictional force is equal to the change in the skier's kinetic energy, given by:
work = change in kinetic energy
Since the skier's final velocity is zero at the top of the slope, the work done by friction is equal to the negative of the initial kinetic energy:
work = - (1/2) * m * v^2
where v is the initial velocity of the skier.
Next, let's consider the uphill part of the skier's motion. As the skier travels up the slope, two forces act on her: the component of the skier's weight parallel to the slope (mg*sinθ), and the force of kinetic friction in the opposite direction (-μmg*cosθ). Here, θ represents the angle of the slope.
The net force acting on the skier can be calculated using:
net force = mg * sin(θ) - μ * mg * cos(θ)
Using Newton's second law of motion, the net force is equal to the mass of the skier multiplied by her acceleration:
net force = m * acceleration
Given that the acceleration is negative (opposite to the direction of motion), we can rewrite this equation as:
mg * sin(θ) - μ * mg * cos(θ) = - m * acceleration
Now, we can use the work-energy principle to relate the work done against this net force to the change in the skier's kinetic energy. The work is given by:
work = change in kinetic energy
Since the skier comes to a stop, the change in kinetic energy is equal to the negative of her initial kinetic energy:
work = - (1/2) * m * v^2
Solving these two equations together, we can find the acceleration as:
mg * sin(θ) - μ * mg * cos(θ) = - (1/2) * m * v^2
Finally, to find the distance traveled up the slope, we can use the kinematic equation:
distance = (v^2 - u^2) / (2 * acceleration)
Substituting the values of v (initial velocity), acceleration, and θ (slope angle), we can solve for the distance traveled up the slope.
Please provide the values of the initial velocity (v), slope angle (θ), and other required values like the coefficient of kinetic friction (μ) and gravity (g), so that we can calculate the distance traveled up the slope.