A rocket in gravity-free empty space has fueled mass \(M_{0}\) and exhaust velocity \(u\) equal to that of a first-stage Saturn V rocket (as used in sending men to the moon): \(M_{0}=3100\) tons \(=28 \times 10^{6} \mathrm{~kg}\) and \(u=2500 \mathrm{~m} / \mathrm{s}\). The ship's acceleration is kept constant at \(10 \mathrm{~m} / \mathrm{s}^{2}\).

(a) Find the initial rate of fuel ejection \(|d M / d t|_{t=0}\).

(b) After how many minutes will the ship mass be reduced to \(1 / e\) of its initial value?

(c) Suppose the ship accelerates as described for 20 minutes. What percent of its initial mass is left? How many kilograms is this? What is the ship's velocity at this time?