Consider the setup in Problem 16, but now let the sand leak at a rate proportional to the bucket’s acceleration. That is, dm/dt = bx. Note that x is negative, so dm is also.
(a) Find the mass as a function of time, m(t).
(b) Find v(t) and x(t) for the times when the bucket contains a nonzero amount of sand. Also find v(m) and x(m). What is the speed right before all the sand leaves the bucket (assuming it hasn’t hit the wall yet)?
(c) What is the maximum value of the bucket’s kinetic energy, assuming it is achieved before it hits the wall?
(d) What is the maximum value of the magnitude of the bucket’s momentum, assuming it is achieved before it hits the wall?
(e) For what value of b does the bucket become empty right when it hits the wall?