A satellite moving with velocity v in a force | vedclass

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Question

(B) According to Newton's second law,the net external force $F$ acting on a system is equal to the rate of change of momentum: $F = \frac{d}{dt}(Mv)$.

Vedclass · Accepted Answer

$-\frac{\alpha v^2}{M}$

Vedclass · Answer

(B) According to Newton's second law,the net external force $F$ acting on a system is equal to the rate of change of momentum: $F = \frac{d}{dt}(Mv)$.Since the satellite is moving in force-free space,the external force $F = 0$.Therefore,$\frac{d}{dt}(Mv) = 0$.Expanding this using the product rule: $M \frac{dv}{dt} + v \frac{dM}{dt} = 0$.We are given the rate of mass collection as $\frac{dM}{dt} = \alpha v$.Substituting this into the equation: $M \frac{dv}{dt} + v(\alpha v) = 0$.Rearranging to solve for acceleration $a = \frac{dv}{dt}$:$M a + \alpha v^2 = 0$.$M a = -\alpha v^2$.$a = -\frac{\alpha v^2}{M}$.

$A$ satellite moving with velocity $v$ in a force-free space collects stationary interplanetary dust at a rate of $\frac{dM}{dt} = \alpha v$,where $M$ is the mass (of satellite + dust) at that instant. The instantaneous acceleration of the satellite is:

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