Question

A rocket of mass $120kg$ is fired in a gravity free space is ejecting gases with velocity $600m/s$ at the rate of $1kg/s$. What will be the initial acceleration of the rocket?

• A. $1m/s^2$
• B. $5m/s^2$
• C. $10m/s^2$
• D. $15m/s^2$

Answer 1

Answer: (B)

$5m/s^2$

Here,

$M=120kg$

$u=600m/s$

$\frac{dm}{dt}=-1kg/s$

Negative because mass is decreasing with time

From Rocket propulsion analysis we have

$mdv=-udm$

$⟹\frac{dv}{dm}=-\frac{u}{m}$ ........(1)

We know from Momentum equation

Let $m$ be the mass of the rocket at an instant and $v$ its velocity at instant

$F=\frac{d}{dt}\left(mv\right)$

$=v\left(\frac{dm}{dt}\right)+m\left(\frac{dv}{dt}\right)$

Using chain rule

$\frac{dv}{dt}=\left(\frac{dv}{dm}\right)\left(\frac{dm}{dt}\right)$

$F=v\left(\frac{dm}{dt}\right)+m\left(\frac{dv}{dm}\right)\left(\frac{dm}{dt}\right)$

Using equation (1)

$=v\left(\frac{dm}{dt}\right)+m(-\frac{u}{m})\left(\frac{dm}{dt}\right)$

$=v\left(1\right)–u(-1)$

$=v+u$

At the beginning velocity of the rocket was $0$,

$v=0$

$F=u=600$

So initial thrust $=600N$

Initial acceleration $=\frac{600}{120}=5m{s}^{-2}$