Question

A space-ship travelling in the interstellar medium is picking up dust at a rate $\frac{d M}{d t}=\alpha v$ , where $\alpha $ is a positive constant, $v$ is the instantaneous velocity and $M$ is the total mass of the space-ship at any instant. The instantaneous acceleration of the space-ship is

• A. $\frac{- 2 \alpha v^{2}}{M}$
• B. $\frac{- \alpha v^{2}}{M}$
• C. $\frac{- \alpha v^{2}}{2 M}$
• D. $- \alpha v^{2}$

Answer 1

Answer: (B)

$\frac{d M}{d t}=\alpha v$
$F_{thrust}=-v\frac{d M}{d t}=Ma$
$\Rightarrow -v\left(\alpha v\right)=Ma$
$\Rightarrow a=-\frac{\alpha v^{2}}{M}$