Problem 44

 A rocket accelerates by burning its onboard fuel, so its mass decreases with time. Suppose the initial mass of the rocket at liftoff (including its fuel) is    , the fuel is consumed at rate r, and the exhaust gases are ejected with constant velocity    (relative to the rocket). A model for the velocity of the rocket at time t is given by the equation

  

where    is the acceleration due to gravity and t is not too large. If

Step 1.

Since velocity is the rate of change of height, we can use part 2 of the fundamental theorem of calculus, to write

 

Since the initial height  , we can write

 

The above equation gives us the height of the rocket at $a$ seconds after launch.