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