A rocket is fired in a gravity-free environment in deep space. The rocket has an initial mass of 8000 kg and ejects gas with a velocity of 2000 m/s. How much gas must be ejected in the first second to give the rocket an acceleration of 30.0 m/s²? A) 120 kg B) 7.50 kg C) 15.0 kg D) 12.0 kg E) 150 kg

Question

Quizplus · Accepted Answer

Explanation:The force on the rocket is equal to the rate of change of momentum of the gas:$$F = \frac{d(mv)}{dt}$$where m is the mass of the gas and v is its velocity. The acceleration of the rocket is:$$a = \frac{F}{m_{rocket}}$$where $m_{rocket}$ is the mass of the rocket. Substituting the first equation into the second equation, we get:$$a = \frac{d(mv)}{dt} \frac{1}{m_{rocket}}$$$$a = \frac{v}{m_{rocket}} \frac{dm}{dt}$$Solving for $\frac{dm}{dt}$, we get:$$\frac{dm}{dt} = \frac{am_{rocket}}{v}$$Substituting the given values, we get:$$\frac{dm}{dt} = \frac{(30.0 \ m/s^2)(8000 \ kg)}{2000 \ m/s} = 120 \ kg/s$$Therefore, the correct answer is A.

A Rocket Is Fired in a Gravity-Free Environment in Deep