A skier traveling 11.0 m/s reaches the foot of a steady upward 19° incline and
glides 15 m up along this slope before coming to rest. What was the average
coefficient of friction? (use conservation of energy method or work energy
theorem)
8. A skier traveling 11.0 m/s reaches the foot of a steady upward 19° incline and glides 15 m up along this slope before coming to rest. What was the average coefficient of friction? (use conservation of energy method or work energy theorem)
Oh, skiing uphill, the perfect way to ruin a downhill sport! Anyway, let's get down to some calculations.
To start off, we'll use the conservation of energy method. The initial kinetic energy of the skier is equal to the work done against friction and the change in gravitational potential energy.
Since the skier comes to rest, their final kinetic energy is zero. Let's call the mass of the skier "m" to keep things simple.
The initial kinetic energy (KE_i) is given by KE_i = (1/2) * m * (11.0 m/s)^2.
The change in gravitational potential energy (ΔPE) is given by ΔPE = m * g * Δh, where g is the acceleration due to gravity (9.8 m/s^2) and Δh is the change in height (15 m * sin(19°)).
The work done against friction (W_f) is equal to the average coefficient of friction (μ) multiplied by the normal force (N) times the distance traveled (d). Since the skier is moving up the slope, the normal force is equal to the weight of the skier, N = m * g.
Putting it all together, we have:
KE_i = W_f + ΔPE
(1/2) * m * (11.0 m/s)^2 = μ * m * g * d + m * g * Δh
Now, let's solve for the average coefficient of friction (μ):
μ = [(1/2) * m * (11.0 m/s)^2 - m * g * Δh] / (m * g * d)
Now, cancel out the 'm' to make the equation simpler:
μ = [(1/2) * (11.0 m/s)^2 - g * Δh] / (g * d)
Plug in the values and do the math, and voila! You'll have the average coefficient of friction. Just remember, I may be a clown, but I'm not clowning around with these calculations! Good luck!
To find the average coefficient of friction, we can use the work-energy theorem. According to the theorem, the net work done on an object is equal to the change in its kinetic energy.
First, let's calculate the initial kinetic energy of the skier. Given that the skier is traveling at 11.0 m/s, the initial kinetic energy (KEi) can be calculated using the formula:
KEi = (1/2) * mass * (velocity)^2
However, we are not given the mass of the skier. Fortunately, we can cancel out the mass by calculating the ratio of the final kinetic energy to the initial kinetic energy, and the masses will cancel out in the equation.
Next, let's calculate the final kinetic energy of the skier. The skier glides up along a slope, so the final kinetic energy (KEf) is zero because the skier comes to a stop.
Now, let's calculate the gravitational potential energy (PE) gained by the skier as it moves up the slope. The formula for gravitational potential energy is:
PE = mass * g * height
Here, g is the acceleration due to gravity (9.8 m/s^2) and height is the vertical distance moved up the slope (15 m).
Since the skier is moving up the slope, the work done by the frictional force is negative because the force and displacement are in opposite directions. The work done by the frictional force is given by:
Work = force * distance * cos(angle)
Here, the force can be calculated using the equation:
force = frictional force = coefficient of friction * normal force
The normal force can be defined as:
normal force = mass * g * cos(theta)
where theta is the angle of inclination of the slope (19°).
Now, using the work-energy theorem:
Work + PE + KEf = KEi
Since KEf is zero in this case:
Work + PE = KEi
Now, we have all the necessary information to calculate the average coefficient of friction using the above equations. Plug in the given values and solve for the unknown coefficient of friction.