Question:
A spaceship in space sweeps stationary interplanetary dust. As a result, its mass increases at a rate $\frac{d M(t)}{d t}=b v^{2}(t)$, where $v(t)$ is its instantaneous velocity. The instantaneous acceleration of the satellite is :
Correct Option: , 2
Solution:
(2) From the Newton's second law,
$F=\frac{d p}{d t}=\frac{d(m v)}{d t}=v\left(\frac{d m}{d t}\right)$ ...(1)
We have given, $\frac{d M(t)}{d t}=b v^{2}(t)$ ...(2)
Thrust on the satellite,
$F=-v\left(\frac{d m}{d t}\right)=-v\left(b v^{2}\right)=-b v^{3}$ [Using (i) and (ii)]
$\Rightarrow F=M(t) a=-b v^{3} \Rightarrow a=\frac{-b v^{3}}{M(t)}$
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