Past exam of the mathematics course of the University of Cambridge / 2021 / ia / Paper 4 / 3C (Dynamics and Relativity)

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- Solution 3C

Solution

Let

M (t) = m_{0} - α t

be the trolley's remaining mass. During

d t

, the expelled mass

- d M

has ground-frame velocity

v + u

. Conservation of momentum gives, to first order,

M v = (M + d M) (v + d v) + (- d M) (v + u),

and hence

M d v = u d M

. Integration yields the rocket equation

v - v_{0} = u lo g \frac{M}{m _{0}} .

At the stopping time,

M = m_{0} e^{- v_{0} / u}

. Therefore the time spent ejecting gas is

\frac{m _{0}}{α} (1 - e^{- v_{0} / u}) .

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