Find the altitude attended by the payload when launched vertically.

The propellant in a single stage sounding rocket occupies 60 % of its initial mass.If all of it is expanded 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 9.91\(\;m/s^{2}\).

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    Using the ideal rocket equation \[V=u_{e}ln\frac{m_{i}}{m_{f}}
    \\\Rightarrow 3000\times ln\left ( \frac{m}{0.4m} \right )
    \\\Rightarrow V=3000\times ln\left ( 2.5 \right )
    \\\Rightarrow V=2748.87\;m/s\]

    Altitude attended by the payload \[\frac{v^{2}}{2g}=\frac{2748.87^{2}}{2\times 9.91}
    \\\approx 381\;km\]

    Answered on 8th October 2019.
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