Question

A rocket is moving at a constant speed in space by burning its fuel and ejecting out the burnt gases through a nozzle. Is there any force acting on the rocket? If yes, how much?
(A) Yes, equal to rate of change in momentum
(B) No
(C) Yes, equal to rate of change in velocity
(D) Yes, equal to change in mass

Answer 1

Hint: The rocket is moving forward in space by ejecting out the burnt gases through a nozzle, as you can see that it is moving forward at a constant speed with the help of fuel injection. Now, you have to consider Newton’s laws of motion in order to comment on the situation and find the magnitude of force acting on the rocket if any.

Complete answer:
Consider a rocket in space moving forward with a constant speed, say having a magnitude of $u$. The fuel is continuously burnt and is ejected through the nozzle. As you can see, that when the fuel gets out of the rocket, the mass of the rocket will definitely change. If we denote the momentum of the rocket as $p_R$, it will be given as $p_R=mu$. You can see that the momentum of the rocket will keep on changing due to the change in mass of the rocket as fuel keeps on ejecting out.

Considering Newton’s 2nd law of motion, it says that the force on an object is equal to the rate of change of momentum of the object with respect to time. Mathematically $F={\displaystyle \frac{dp}{dt}}$ $....\left(1\right)$.
So, if there is a force acting on the rocket, it should come out to be non-zero from equation $\left(1\right)$. Let us put the value of momentum of the rocket in equation $\left(1\right)$.
Then, $F_R={\displaystyle \frac{d{p}_R}{dt}}$
$$F_R={\displaystyle \frac{d\left(mu\right)}{dt}}$$.

Apply product rule of differentiation. $$F_R=m{\displaystyle \frac{du}{dt}}+u{\displaystyle \frac{dm}{dt}}$$, as $u$ is constant, its derivative will be zero and as the mass keeps on changing due to ejection of fuel, the derivative of mass with respect to time will be non-zero. Finally, you are left with $F_R=u{\displaystyle \frac{dm}{dt}}$, which is a non-zero quantity. You can take the speed inside and write it as momentum.

So, yes, there is a force equal to change in momentum acting on the rocket and this force is applied by the burnt gases which are continuously ejected through the nozzle, or say the thrust applied by the gases. Therefore, if a rocket is moving at a constant speed in space by burning its fuel and ejecting the burnt gases through a nozzle, yes, there is a force acting on the rocket and is equal to rate of change in momentum of the rocket.

Hence, option A is correct.

Note: Keep in mind Newton's Laws of motion, it helps you to solve any question related to mechanics. For questions of this type, remember that the force on any object is given by the rate of change of momentum with respect to time. Here, you should note that if you consider your system as a fuel + rocket, then the force on the whole system is zero if you are considering the system is isolated.