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NEET PHYSICSMedium

A satellite in force-free space sweeps stationary interplanetary dust at a rate 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?

A

−2\alpha v²/M

B

−\alpha v²/M

C

\alpha v²/M

D

−\alpha v²

Step-by-Step Solution

  1. Apply Newton's Second Law for Variable Mass: The external force on the system is the rate of change of momentum (Fext=dpdtF_{ext} = \frac{dp}{dt}). In force-free space, Fext=0F_{ext} = 0.
  2. Expand the Derivative: P=MvP = Mv. Fext=d(Mv)dt=Mdvdt+vdMdt=0F_{ext} = \frac{d(Mv)}{dt} = M\frac{dv}{dt} + v\frac{dM}{dt} = 0 [Source 60]
  3. Substitute Given Rate: We are given the rate of mass accumulation \frac{dM}{dt} = lpha v. M\frac{dv}{dt} + v(lpha v) = 0 M a + lpha v^2 = 0
  4. Solve for Acceleration (a): a = -\frac{lpha v^2}{M} The negative sign indicates the satellite is slowing down (retardation). The deceleration (magnitude of negative acceleration) is \frac{lpha v^2}{M}.
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