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A satellite in a force free space sweeps stationary interplanetary dust at a rate $(dM/dt)=\alpha v$. The acceleration of the satellite is-
A. $\dfrac{{ - 2\alpha {v^2}}}{M}$
B. $\dfrac{{ - \alpha {v^2}}}{M}$
C. $\dfrac{{ - \alpha {v^2}}}{{2M}}$
D. $ - \alpha {v^2}$

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Last updated date: 18th Jun 2024
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Answer
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Hint: When the satellite sweeps the dust, its mass increases at the given rate. If there is no external force acting on the satellite the only acceleration will be caused by the slowing due to an increase in the mass of the satellite. It can be calculated using Newton’s second law.

Complete step by step answer:
In the question, it is given that there is no external force acting on the satellite. Therefore the change caused in the motion of the satellite is purely an effect of its increasing mass.
We know that in a system if there is a change in momentum a force would also be applied. It is an application of Newton's 2nd law, where force is defined as the rate of change of momentum.
Here, the momentum of the satellite is given by-
$P = Mv$ where M is the mass of the satellite and v is its velocity.
Any force on the satellite can be given by-
$F = \dfrac{{dP}}{{dt}} = \dfrac{{d(Mv)}}{{dt}}$
Rewriting,
$F = v\dfrac{{dM}}{{dt}} + M\dfrac{{dv}}{{dt}}$
It is given in the question that the rate of change in mass is-
$\dfrac{{dM}}{{dt}} = \alpha v$
And external force is 0,
Putting these values in the equation we get-
$0 = v(\alpha v) + M\left( {\dfrac{{dv}}{{dt}}} \right)$
$M\left( {\dfrac{{dv}}{{dt}}} \right) = - \alpha {v^2}$
$\left( {\dfrac{{dv}}{{dt}}} \right) = \dfrac{{ - \alpha {v^2}}}{M}$

So, the correct answer is “Option B”.

Note:
Although the object doesn’t experience any external force, it still undergoes a deceleration because of the interaction between the mass present outside the system. As force is defined as a change in momentum, any change in either mass or velocity would result in the application of force. This addition of mass decreases the total energy of the satellite causing it to slow down.