Two blocks, A and B, of the same mass are on a fixed inclined plane, which makes a 30^{\circ} angle with the horizontal. At time t=0, A is a distance \ell=5 \mathrm{~cm} along the incline above B, and both blocks are at rest. Suppose the coefficients of static and kinetic friction between the blocks and the incline are
and that the blocks collide perfectly elastically. Let v_{A}(t) and v_{B}(t) be the speeds of the blocks down the incline. For this problem, use g=10 \mathrm{~m} / \mathrm{s}^{2}, assume both blocks stay on the incline for the entire time, and neglect the sizes of the blocks.
a. Graph the functions v_{A}(t) and v_{B}(t) for t from 0 to 1 second on the provided answer sheet, with a solid and dashed line respectively. Mark the times at which collisions occur.
b. Derive an expression for the total distance block A has moved from its original position right after its n^{\text {th }} collision, in terms of \ell and n.
Now suppose that the coefficient of block B is instead \mu_{B}=\sqrt{3} / 2, while \mu_{A}=\sqrt{3} / 6 remains the same.
c. Again, graph the functions v_{A}(t) and v_{B}(t) for t from 0 to 1 second on the provided answer sheet, with a solid and dashed line respectively. Mark the times at which collisions occur.
d. At time t=1 \mathrm{~s}, how far has block A moved from its original position?
Following are the answer sheets for the graphing portion of the question.
A1: Collision Course
(a)
(c)
