Fig. 1

Shown in Fig. 1 is an experimental car propelled by a rocket motor. The drag force (air resistance) is given by . The initial mass of the car, which includes fuel of mass , is . The rocket motor is burning fuel at the rate of with an exhaust velocity of relative to the car. The car is at rest at . Show that the velocity of the car is given by, for ,

,

where , and is the time when the fuel is burnt out.

We have derived the governing equation of rocket flight in “Viva Rocketry! Part 1“, namely,

From , we have

.

Apply air resistance as the external force, (1) becomes

.

And the car is at rest initially implies

.

It follows that the motion of the car can be modeled by an initial-value problem

It suffices to show that the given is the solution to this initial-value problem:

Fig. 2

An alternative is obtaining the stated through solving (2).

Fig. 3

The fact that simplifies the result considerably,

Divide both the numerator and denominator of (3) by then yields

which is equivalent to the given since .

At time , the fuel is burnt out. It means

.

Therefore,

Exercise 1: Solve (2) manually.

*Hint*: The differential equation of (2) can be written as .

Exercise 2: For , what is the burnout velocity of the car?