Discussion
The propellant in a single stage sounding rocket occupies 60% of its initial mass. If all of it is expended instantaneously at an equivalent exhaust velocity of 3000 m/s, what would be the altitude attained by the payload when launched vertically?
[Neglect drag and assume acceleration due to gravity to be constant at 9.81 m/s2.]
Answer: D
The payload reaches about 385 km.
With all propellant burnt instantaneously there are no gravity losses during the burn, so the full rocket-equation velocity is reached at once. The mass ratio is \(m_{0}/m_{f} = 1/0.4 = 2.5\):
\[ \begin{aligned} \Delta V &= V_{e}\ln\frac{m_{0}}{m_{f}} = 3000\ln 2.5 \\ &= 3000 \times 0.9163 = 2749\ \text{m/s} \end{aligned} \]
The payload then coasts upward against constant gravity with no drag:
\[ \begin{aligned} h &= \frac{\Delta V^{2}}{2g} = \frac{2749^{2}}{2 \times 9.81} \\ &= 3.85 \times 10^{5}\ \text{m} \end{aligned} \]
which is 385 km. The smaller options would need a lower burnout velocity, for example from using a mass ratio below 2.5, which the 60% propellant fraction does not allow.
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