As a car moves, it burns fuel to overcome resistance and drive the car forward. We can apply Newton's second law of motion to find the equation of motion of the car.

If the rocket engine burns up fuel at a constant ratekg/s, expelling the burnt product at a speedm/s relative to the car, and if the engine is turned on atthen atthe mass of the car iswhereis the initial mass of the car.

The rate of change of momentum of the expelled gases isand according to Newton's third law of motion this will result in an equal reaction force on the car.

If we assume that the total resistance to motion is proportional to the velocityof the car, then applying Newton's second law gives

Sincewe can write

Separating variables givesIfwhenthen

Exponentiation givesand further rearrangement and simplification gives