Problem 8.43

8.43. a)

We know from the chapter that the velocity of the rocket as a function of the exhaust velocity and the mass of the fuel and rocket is given by

$v = v_{e} \ln \frac{M_{i}}{M_{f}}$

rearranging this yields

M_i  =  M_f e^v/v_e

We know the ratio of the velocity of the exhaust to that of the rocket is 5. Solving for the initial mass we find

$M_{i} = e^{5} (3.0 \times 10^{3} kg) ~=~ 4.45 \times 10^{5} kg$

Solving for the mass of the fuel and oxidizer, we get

$\Delta M ~=~ M_{i} - M_{f} ~=~ (445 - 3) \times 10^{3} kg ~=~ 442$ metric tons.

b)

Solving again for the mass of the fuel and oxidizer, we find that

$\Delta M ~=~ (3 e^{2} - 3) \times 10^{3} kg ~=~ 19.2$ metric tons.

Note that with the exponential dependence, a small difference in the engine efficiency makes a large differece in the amount of fuel and oxidizer required.