Question

A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{dM\left(t\right)}{dt}=b{v}^2\left(t\right)$, where $v\left(t\right)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is:

• A. $–b{v}^3\left(t\right)$
• B. $–b{v}^3/M\left(t\right)$
• C. $–2b{v}^3/M\left(t\right)$
• D. $–b{v}^3/2M\left(t\right)$

Answer 1

The correct option is B

$–b{v}^3/M\left(t\right)$

Step 1: Given data

Mass, $\frac{dM\left(t\right)}{dt}=b{v}^2\left(t\right)$

Thrust force, $T=M\times a$

Step 2: Find the instantaneous acceleration of the satellite

There is no external force on the rocket, it is just the thrust force. So, it is given by

$\begin{array}{rcl}T& =& \frac{dM\left(t\right)}{dt}\left(v_{rel}\right)\\ Ma& =& [–b{v}^2(t\left)\right]v\\ a& =& –b{v}^3/M\left(t\right)\end{array}$

Hence, option B is correct.