No new concepts or techniques are involved here, and the main function of this homework is to refamiliarize you with the mechanics concepts you have already met as an undergraduate.

I-A:      Goldstein exercise 1-2.

I-B:      Goldstein exercise 1-3.
Hint: To get started, assume that the rocket has mass m at a particular instant part-way into its burn, at which instant it is moving upward with velocity v. Thus, the momentum of this system relative to the earth is mv. At a time dt later, how has the momentum of this system changed?

The rocket has burned an extra amount of fuel dm. Its mass has changed by this amount, and it is faster by dv. Don't forget that, at the end of dt, the fuel mass dm has become hot gas streaming downward at velocity v'- v relative to the earth, and must be included in the total momentum of the system under consideration.

You need to be careful with the sign of dm. The best approach is to write the rocket mass at the end of dt as m + dm, but keep in mind that dm/dt for the rocket is a negative number as the fuel burns.

I-C:      Goldstein exercise 1-4.

I-D:      Goldstein's problem 1-3 also appeared in his first edition, published in 1950. In that edition, the only difference is he points out that the numerical values in the problem refer to the V-2 rocket, which the Germans used against southeastern England towards the end of World War II. At the time of writing, that was the world's most technologically advanced rocket. Its empty weight was 3 tonnes and it carried 9 tonnes of fuel and payload (explosive). Therefore, its ratio of fuel weight to empty weight was vastly less than what would be needed to reach earth escape velocity. Nevertheless, modern rockets do not have dramatically better performance than the V-2, and exploration of Mars and other planets is possible only by using multistage rockets.

(a) If the speed of a rocket (relative to the earth) were to reach and then exceed the speed v' of its exhaust gas (relative to the rocket), then viewed from the earth, its exhaust would change direction and would actually be moving towards the rocket. Is such a situation possible, or is there a flaw in this reasoning?

(b) Imagine that you are back in 1944 using V-2 technology, and wish to explore space. You use a V-2 as your first stage. The explosive warhead is replaced by the upper stages and final payload. The empty weight of each successive stage is one-tenth of the previous one, but otherwise all upper stages have the same performance as the V-2. For example, this means that the mass of the empty second stage (no payload or fuel), is 0.3 tonnes, and the total mass of its payload and fuel is 0.9 tonnes. Assume a final payload of about 11% of the total mass of the last stage. Thus, the final payload would be 132 kg for a two-stage rocket. How many stages would be needed to leave the earth travelling straight up, and what final payload could be carried?