A space-ship travelling in the interstellar medium is picking up dust at a rate (d M/d t)=α v , where α 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

Question

A space-ship travelling in the interstellar medium is picking up dust at a rate (d M/d t)=α v , where α 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) (- 2 α v2/M) (B) (- α v2/M) (C) (- α v2/2 M) (D) - α v2

Tardigrade Admin · Accepted Answer

(d M/d t)=α v Fthrust=-v(d M/d t)=Ma ⇒ -v(α v)=Ma ⇒ a=-(α v2/M)

Q. A space-ship travelling in the interstellar medium is picking up dust at a rate $\frac{d M}{d t} = αv$ , where $α$ 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

$\frac{- 2 α v ^{2}}{M}$

$\frac{- α v ^{2}}{M}$

$\frac{- α v ^{2}}{2 M}$

$- α v^{2}$

$\frac{d M}{d t} = αv$
$F_{t h r u s t} = - v \frac{d M}{d t} = M a$
$\Rightarrow - v (αv) = M a$
$\Rightarrow a = - \frac{α v ^{2}}{M}$

Q. A space-ship travelling in the interstellar medium is picking up dust at a rate dtdM​=αv , where α 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

M−2αv2​

M−αv2​

2M−αv2​

−αv2