The initial mass of a rocket is 1000 kg. Calculate at what rate the fuel should be burnt so that the rocket is given an acceleration of 20 m s$$^{-2}$$. The gases come out at a relative speed of 500 m s$$^{-1}$$, with respect to the rocket: [Use $$g = 10$$ m s$$^{-2}$$]

We consider the instantaneous motion of the rocket when its mass is still $$m = 1000\text{ kg}$$ and it has to acquire an upward acceleration of $$a = 20\ \text{m s}^{-2}$$. At that very instant the rocket ejects gases backward with a speed, relative to the rocket, of $$u = 500\ \text{m s}^{-1}$$.

According to the variable-mass (rocket) mechanics, the backward ejection of mass produces a forward thrust. The magnitude of this thrust is obtained from the basic relation

$$\text{Thrust} = \left(\dfrac{dm}{dt}\right)u,$$

where $$\dfrac{dm}{dt}$$ is the rate at which mass is being lost by the rocket (this rate is positive in magnitude although the actual change in mass is negative).

The forces acting on the rocket, all taken along the vertical line of motion, are:

1. Upward thrust $$\left(\dfrac{dm}{dt}\right)u$$,

2. Downward weight $$mg$$.

The net upward force must produce the required upward acceleration. Hence Newton’s second law gives

$$\left(\dfrac{dm}{dt}\right)u - mg = ma.$$

Now we substitute the known numerical values. We have

$$m = 1000\ \text{kg}, \qquad g = 10\ \text{m s}^{-2}, \qquad a = 20\ \text{m s}^{-2}, \qquad u = 500\ \text{m s}^{-1}.$$

Putting these into the above equation,

$$\left(\dfrac{dm}{dt}\right)(500) - (1000)(10) = (1000)(20).$$

First evaluate the right-hand side:

$$ (1000)(20) = 20000. $$

So we have

$$500\dfrac{dm}{dt} - 10000 = 20000.$$

Move the weight term to the right:

$$500\dfrac{dm}{dt} = 20000 + 10000 = 30000.$$

Solve for $$\dfrac{dm}{dt}$$ by dividing both sides by 500:

$$\dfrac{dm}{dt} = \dfrac{30000}{500}.$$

Carry out the division:

$$\dfrac{dm}{dt} = 60\ \text{kg s}^{-1}.$$

Thus the fuel must be burnt at a rate of $$60\ \text{kilograms per second}$$ to give the rocket the desired acceleration.

Hence, the correct answer is Option B.