Question

What is the equation of motion for the rocket, if $m$ is the mass of the rocket at a given moment, $\overrightarrow{w}$ is its acceleration, and $\overrightarrow{F}$ is the external force.

• A. $m\overrightarrow{w}=\overrightarrow{F}-2\mu \overrightarrow{u}$
• B. $m\overrightarrow{w}=\overrightarrow{F}+\mu \overrightarrow{u}$
• C. $m\overrightarrow{w}=\overrightarrow{F}-3\mu \overrightarrow{u}$
• D. $m\overrightarrow{w}=\overrightarrow{F}-\mu \overrightarrow{u}$

Answer 1

The correct option is D $m\overrightarrow{w}=\overrightarrow{F}-\mu \overrightarrow{u}$

Let, at time instant $t$, its mass be $m$ and velocity be $\overrightarrow{v}$

Then after small time $\Delta t$, it mass will be $m-\Delta m$ and velocity will be $\overrightarrow{v}+\overrightarrow{\Delta v}$

$\therefore \overrightarrow{F}=\frac{P_f-P_i}{\Delta t}=\frac{[(m-\Delta m)(\overrightarrow{v}+\overrightarrow{\Delta v})+\left(\Delta m\right)(\overrightarrow{v}+\overrightarrow{u})]-m\overrightarrow{v}}{\Delta t}$, neglecting $\Delta m\overrightarrow{\Delta v}$ as it is very small ,

$\Rightarrow$$\overrightarrow{F}=\frac{m\overrightarrow{\Delta v}+\Delta m\overrightarrow{u}}{\Delta t}=m\left(\frac{\overrightarrow{\Delta v}}{\Delta t}\right)+\left(\frac{\Delta m}{\Delta t}\right)\overrightarrow{u}=m\overrightarrow{\omega }+\mu \overrightarrow{u}$

$\Rightarrow$ $m\overrightarrow{\omega }=\overrightarrow{F}-\mu \overrightarrow{u}$