A rocket sled moves along the horizontal plane under the presence of a friction force mg, where

A rocket sled moves along the horizontal plane under the presence of a friction force μmg, where m is the mass of the sled at that moment and μ is the coefficient of kinetic friction. The rocket propels itself by ejecting mass at a constant rate dm/dt = −R (R is a positive number, because the sled’s mass is decreasing with time), and the fuel is ejected at a constant speed u relative to the sled. The sled starts from rest with initial mass M, and stops ejecting fuel when half the mass has been expended.

1. a) How long does it take for the sled to finish ejecting its fuel?

2. b) Write an expression for the mass of the sled as a function of time.

3. c) Using any method you would like, derive the following differential equation governing the motion of the sled: (M−Rt)dv =−(M−Rt)gμ+Ru

4. d) Separate variables and integrate to find v(t), the velocity of the sled as a function of time.

5. e) After the rocket stops firing, the sled continues to coast an additional distance d before coming to a stop due to friction. Show that this extra distance is d = (2^2/2μg) (ln2 - μMg/2Ru)^2

6. Show that the total time of the sled’s journey (including both the propulsion stage and the period of coasting) is independent of the rate R at which the fuel is ejected. Find this time, in terms of u, g and μ.