A rocket with an initial mass of 1000 kg adjusts its thrust by varying the rate at which mass is ejected. The ejection speed relative to the rocket is 40 km/s. If the acceleration of the rocket is to have a magnitude of 20 m/s² at an instant when its mass is 80% of the original mass, at what rate is mass being ejected at that instant? Ignore any external forces on the rocket. A) 0.40 kg/s B) 0.50 kg/s C) 0.60 kg/s D) 0.70 kg/s E) 0.80 kg/s

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Quizplus · Accepted Answer

The correct answer is A.Explanation:The rate of change of momentum of the rocket is equal to the net force acting on it. The net force is the thrust of the rocket, which is equal to the rate of change of mass times the ejection speed. The acceleration of the rocket is equal to the net force divided by the mass of the rocket.At the instant when the mass of the rocket is 80% of the original mass, the acceleration of the rocket is 20 m/s^2. The ejection speed is 40 km/s, which is equal to 40,000 m/s. The mass of the rocket is 800 kg.The rate of change of mass is equal to the thrust divided by the ejection speed. The thrust is equal to the mass of the rocket times the acceleration of the rocket.Therefore, the rate of change of mass is:$$\frac{dm}{dt} = \frac{ma}{v_e}$$$$\frac{dm}{dt} = \frac{(800 	ext{ kg})(20 	ext{ m/s}^2)}{40,000 	ext{ m/s}}$$$$\frac{dm}{dt} = 0.40 	ext{ kg/s}$$

A Rocket with an Initial Mass of 1000 Kg Adjusts