Question

A 5000 kg rocket is set for vertical firing the exhaust speed is $800m/s$ to give an initial acceleration of $20m/s^2$, the amount of gas ejected per second to supply. The needed thrust will be:
a. $137.5kg/s$.
b. $185.5kg/s$.
c. $187.5kg/s$.
d. $127.5kg/s$.

Answer 1

Propulsion system is a machine which is used to produce thrust so that the machine can move forward. The rocket has a propulsion system which burns the fuel and produces thrust which lifts the rocket from the ground.

Formula used:
The relation of the rocket propulsion is given by,
$\Rightarrow \left( {\dfrac{{\Delta m}}{{\Delta t}}} \right)v - mg = ma$
Where kg per sec is $\dfrac{{\Delta m}}{{\Delta t}}$, the velocity of the exhaust fuel is v, the acceleration of the rocket is a and the acceleration due to gravity is g.

Complete step by step answer:
It is given the problem that the $5000kg$ rocket is set for vertical firing. The exhaust speed is $800m/s$ to give an initial acceleration of $20m/s^2$, we need to find the thrust of the rocket.

Step1:
From the rocket propulsion equation we have,
$\Rightarrow \left( {\dfrac{{\Delta m}}{{\Delta t}}} \right)v - mg = ma$
Where $\dfrac{{\Delta m}}{{\Delta t}} =$ mass of gas ejected per sec.
m = Mass of the rocket $= 5000kg$
v = Speed of exhaust gases $= 800m/s$
a = Initial acceleration $= 20m/s^2$

Step2:
Now substitute all the values in the above equation and calculate the gas ejected per second.
$\Rightarrow \left( {\dfrac{{\Delta m}}{{\Delta t}}} \right)v = ma + mg$
$\Rightarrow \left( {\dfrac{{\Delta m}}{{\Delta t}}} \right)v = \left( {a + g} \right)m$
The acceleration due to gravity is $g = 10m/s^2$.
$\Rightarrow \left( {\dfrac{{\Delta m}}{{\Delta t}}} \right) \times 800 = \left( {20 + 10} \right) \times 5000$
On solving we get.
$\Rightarrow \left( {\dfrac{{\Delta m}}{{\Delta t}}} \right) = \dfrac{{30 \times 5000}}{{800}} = 187 \cdot 5kg/s$
The thrust of the rocket is equal to $187 \cdot 5kg/s$.

Hence, the correct answer is option (C).

Note: It is advised to students to understand and remember the formula of the rocket propulsion as it is very useful in solving these problems. The thrust is the force which is perpendicular to the surface. According to Newton's third law of motion, every action has an equal and opposite reaction the rocket takes lifts from the ground.