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 rate
kg/s, expelling the burnt product at a speed
m/s relative to the car, and if the engine is turned on at
then at
the mass of the car is
where
is the initial mass of the car.
The rate of change of momentum of the expelled gases is
and 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 velocity
of the car, then applying Newton's second law gives
Since
we can write
Separating variables gives
If
when
then
Exponentiation gives
and further rearrangement and simplification gives