Assignment:
Question1. (a) Consider a rocket traveling in a straight line subject to an external force Fext acting along the same line. Show the equation of motion is
mv = - mvex + Fext. (1)
(b) Specialize to the case of a rocket taking off vertically (from rest) in a gravitational field g, so the equation of motion becomes
mv = - mvex – mg. (2)
Assume that the rocket ejects mass at a constant rate, m = -k (where k is a positive constant), so that m = mo – kt. Solve equation (2) for v as a function of t, using separation of variables (rewriting the equation so that all terms involving v are on the left and all terms involving t on the right).
(c) Using the data the initial mass is 2x106 kg, the final mass (after 2 minutes) is about 1x106 kg, the average speed vex is about 3000m/s, and the initial velocity is zero, find the space shuttle’s speed two minutes into flight, assuming (what is nearly true) that it travels vertically up during the period and that g doesn’t change appreciably. Compare with the corresponding result if there were no gravity.
(d) Describe what would happen to a rocket that was designed so that the first term on the right of Equation (2) was smaller than the initial value of the second.
Question 2. Integrate v(t) from part (b) in the previous problem and show that the rocket’s height as a function of t is
y(t) = vext – (1/2) gt2 – (mvex/k) ln (mo/m).
Using the numbers in problem 1, estimate the space shuttle’s height after two minutes.
Question 3.
(a) We know that the path of a projectile thrown from the ground is a parabola (if we ignore air resistance). In the light of the result Fext = MR. what would be the subsequent path of the CM of the pieces if the projectile exploded in midair?
(b) A shell is fired from level ground so as to hit a target 100m away. Unluckily the shell explodes prematurely and breaks into two equal pieces. The two pieces land at the same time, and one lands 100m beyond the target. Where does the other piece land?
(c) Is the same result true if they land at different times (with one piece still landing 100m beyond the target)?