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The engine of a rocket in outer space, far from any planet, is turned on. The rocket ejects burned fuel at a constant rate; in the first second of firing, it ejects $\frac{1}{120}$ of its initial mass $m_{o}$ at a relative speed of $2400 m / s$ . What is the rocket's initial acceleration (in m / s²)?

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Detailed Solution

IDENTIFY and SET UP: We are given the rocket's exhaust speed $v_{e x}$ and the fraction of the initial mass lost during the first second of firing, from which we can find $d m / d t .$

EXECUT: The initial rate of change of mass is

$\frac{d m}{d t} = - \frac{m_{0} / 120}{1 s} = - \frac{m_{0}}{120 s}$

$α = - \frac{v_{e x}}{m_{0}} \frac{d m}{d t} = - \frac{2400 m / s}{m_{0}} (- \frac{m_{0}}{120 s}) = 20 m / s^{2}$

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