A model car is propelled by a cylinder of carbon dioxide gas.The cylinder emits gas at a rate of 4.5 g/s with an exit speed of 80.0 m/s.The car has a mass of 400 g,including the CO₂ cylinder.Starting from rest,what is the car's initial acceleration? A)0.90 m/s² B)4.5 m/s² C)9.0 m/s² D)36 m/s²

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Quizplus · Accepted Answer

The correct answer is A.The force exerted by the gas on the car is equal to the rate of change of momentum of the gas:$$F = \frac{d(mv)}{dt} = m\frac{dv}{dt} + v\frac{dm}{dt}$$where m is the mass of the gas, v is the exit speed of the gas, and t is time.The rate of change of mass of the gas is equal to the rate at which the gas is emitted from the cylinder:$$\frac{dm}{dt} = -4.5 g/s$$The mass of the car is constant, so the rate of change of momentum of the car is equal to the force exerted by the gas:$$F = m\frac{dv}{dt}$$The acceleration of the car is equal to the force divided by the mass of the car:$$a = \frac{F}{m} = \frac{m\frac{dv}{dt}}{m} = \frac{dv}{dt}$$The initial acceleration of the car is equal to the exit speed of the gas divided by the time it takes for the gas to be emitted from the cylinder:$$a = \frac{v}{t} = \frac{80.0 m/s}{4.5 g/s} = 0.90 m/s^2$$

A Model Car Is Propelled by a Cylinder of Carbon