Description
1. You are in a rocket ship, in outer space. You have a nuclear reactor that supplies a
constant power, P0, and a large supply of iron pellets. The iron pellets comprise 99/100
of your ship’s mass, m.
You can use the power to eject the tiny iron beads out the back
of your ship with an electromagnetic “gun”. You can control the rate at which you
fire them and their velocity, but are limited by your power plant. (You can’t fire an
arbitrarily large mass at an arbitrarily large velocity.)
As you fire off the beads, your
ship moves in the opposite direction to conserve momentum. In addition, the mass of
your ship decreases. (You can solve this using a local constraint, but that’s the hard
way.)
(a) If you use the energy of your reactor over a time ∆t to launch a packet of mass
∆m out the rear of your ship, what is the momentum of this “exhaust” packet
relative to your ship?
(b) Now assume that you fire pellets continuously at a constant rate during the interval 0 < t < tf . If you start from rest, what is your final velocity?
(c) However, you do not have to fire pellets at a constant rate. Find the optimal firing
rate dm/dt in the interval 0 < t < tf so that your final velocity is a maximum
after a time tf , assuming that you started from rest.
(d) What is your final velocity in part (c) ? How does it compare to the answer in
part (b)?
You may find it helpful to review rockets in your favorite Freshman physics book.
2. Consider the functional
I[y(x), y0
(x)] = Z xf
0
∂y
∂x!2
+ α y
∂y
∂x
dx
(a) Find the function y(x) that extremizes I subject to the boundary conditions that
y(0) = 0 and y(xf ) = yf .
(b) How does your answer depend upon α? Why?
3. Byron and Fuller, chapter 2, problem 6.
4. Byron and Fuller, chapter 2, problem 7.
