7. A skier starts at a height h on a downward, frictionless, curved slope, at rest. She goes
down the slope and then travels along a flat horizontal plane for which the coefficient of
kinetic friction is $\mu_k$ until she comes to a stop. The skier has a mass m and the distance
on the flat plane that she travels is x.
a. Find an expression for the speed of the skier when she is at a height of h/2, in
terms of g and h.
b. Find the speed of the skier when she reaches the end of the slope in terms of
g and h.
c. What must $\mu_k$ be so that the skier comes to rest after she travels a distance
x? Write your answer in terms of any variables given in the prompt and
physical constants as needed.
d. Find the amount of time that the skier travels on the flat horizontal plane.
Write your answer in terms of any variables given in the prompt, and physical
constants as needed.
8.
a. Derive an expression for the power required to keep an object in constant
acceleration, if the object starts at rest (do not include v or x in your answer: you
should have a function of a, t, and m). Assume this is a conservative system. Why
should the power increase with acceleration, and why is it non-zero?
b. What is the average power required to keep an object in constant acceleration,
in terms of the same variables as in part a?
9. Consider an object of mass m which starts at the top of an inclined plane with
coefficient of friction $\mu_k$, height h, and angle of elevation $\theta$. How much power is
required by the frictional force as the object slides down the ramp, in terms of m, $\mu_k$, g,
h, $\theta$, and t?
Do the limits for the angle of elevation make sense? Why or why not?
10. Consider a 2 kg object with a speed of 8 m/s. If another object has 3 times the kinetic
energy of the first one and is moving with a speed of 6 m/s, how massive is the second
object?
11. Consider the mass-pulley system below, which starts at rest (no friction).
a. If the mass $m_2$ falls a height h, derive an expression for the speed of mass $m_1$ in
terms of the height, masses, and physical constants as needed.
b. Derive an equation (in terms of h, masses, and physical constants) for the
amount of time that it takes for the mass $m_2$ to fall a distance h if the
acceleration of the system is a = $\frac{m_2}{m_1+m_2}g$.
