Question
A $$5000\,kg$$ rocket is set for vertical firing. The exhaust speed is $$800\,m{s^{ - 1}}.$$ To give an initial upward acceleration of $$20\,m{s^{ - 2}},$$ the amount of gas ejected per second to supply the needed thrust will be
$$\left( {g = 10\,m{s^{ - 2}}} \right)$$
A.
$$127.5\,kg\,{s^{ - 1}}$$
B.
$$187.5\,kg\,{s^{ - 1}}$$
C.
$$185.5\,kg\,{s^{ - 1}}$$
D.
$$137.5\,kg\,{s^{ - 1}}$$
Answer :
$$187.5\,kg\,{s^{ - 1}}$$
Solution :
Given: Mass of rocket $$\left( m \right) = 5000\,kg$$
Exhaust speed $$\left( v \right) = 800\,m/s$$
Acceleration of rocket $$\left( a \right) = 20\,m/{s^2}$$
Gravitational acceleration $$\left( g \right) = 10\,m/{s^2}$$
We know that upward force
$$F = m\left( {g + a} \right) = 5000\left( {10 + 20} \right) = 5000 \times 30 = 150000\,N.$$
We also know that amount of gas ejected
$$\left( {\frac{{dm}}{{dt}}} \right) = \frac{F}{v} = \frac{{150000}}{{800}} = 187.5\,kg/{s^{ - 1}}$$