Let the instant velocity of a rocket be . In the instantaneous rest frame of the rocket, particles of the gas cloud collide with it head-on with the same absolute velocity , and bounce-off with the same velocity. Hence, in the gas frame, they change velocity from zero to . The momentum is thus constantly transferred from the rocket to gas particles, as a result of which, the rocket loses the same amount of momentum. In a small time , the mass of new particles involved in collisions is , where is the cross-sectional area of the rocket, and is the gas mass density. The momentum loss by the rocket is then . If the mass of the rocket is , this should be equal to , where is the change in the rocket velocity. Equating these quantities and dividing by , we obtain a differential equation . Its solution with the initial condition is
Requiring, by the statement of the problem, , we obtain the equation
whence we find the answer:
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