Question

In a rocket of mass $1000kg$fuel is consumed at a rate of $40kg/s$. The velocity of the gases ejected from the rocket is $5 \times {10^4}m/s$. The thrust on the rocket is
A. $2 \times {10^3}N$
B. $2.5m/{s^2}$
C. $2 \times {10^6}N$
D. $2 \times {10^9}N$

Answer 1

The thrust on the rocket can be calculated by simply substituting the values given, in the appropriate formula.
Formula used
${F_t} = u\dfrac{{dm}}{{dt}}$
Where $\dfrac{{dm}}{{dt}}$ is the rate of change of mass with respect to time (mass flow rate of exhaust) and $u$is the speed of the exhaust gases measured relative to the rocket and ${F_t}$ is the thrust generated.

Complete step by step solution
Thrust is a reaction force. According to Newton’s third law of motion, to every action there is an equal and opposite reaction. Thrust comes into consideration when a system expels mass in one direction. This mass causes a force of equal magnitude to act on the system propelling the system into the opposite direction.
Here, we are given a rocket of a fixed mass which is consuming fuel at the rate of $40kg/s$.
That means the mass of the rocket is reducing at the rate of $40kg/s.$
Here, thrust can be calculated by using the formula,
${F_t} = u\dfrac{{dm}}{{dt}}$
Where $\dfrac{{dm}}{{dt}}$ is the rate of change of mass with respect to time (mass flow rate of exhaust) and $u$is the speed of the exhaust gases measured relative to the rocket and ${F_t}$ is the thrust generated.
Therefore,
$\begin{gathered} {F_t} = 5 \times {10^4} \times 40 \\\ \Rightarrow {F_t} = 2 \times {10^6}N \\\ \end{gathered}$

Thus, the correct option is C.

Note: Thrust is a direct consequence of Newton's third law of motion. The power needed to generate thrust and the force of the thrust can be expressed using the non-linear relation ${P^2} \propto {F_t}^3$. However this relation is only valid for those bodies that are at a standstill.