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(C) The force acting on the satellite is given by Newton's second law for a variable mass system: $F = \frac{d}{dt}(Mv)$.Since the satellite is in for

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$ - \alpha v^2/M$

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(C) The force acting on the satellite is given by Newton's second law for a variable mass system: $F = \frac{d}{dt}(Mv)$.Since the satellite is in force-free space,the net external force $F = 0$.Expanding the derivative,we get: $F = M \frac{dv}{dt} + v \frac{dM}{dt} = 0$.Given the rate of mass accumulation is $\frac{dM}{dt} = \alpha v$,we substitute this into the equation:$M \frac{dv}{dt} + v(\alpha v) = 0$.Rearranging to solve for acceleration $a = \frac{dv}{dt}$:$M \frac{dv}{dt} = -\alpha v^2$.Therefore,the deceleration (or acceleration) is $a = -\frac{\alpha v^2}{M}$.

$A$ satellite in force-free space sweeps stationary interplanetary dust at a rate $\frac{dM}{dt} = \alpha v$,where $M$ is the mass,$v$ is the velocity of the satellite,and $\alpha$ is a constant. What is the deceleration of the satellite?

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