A satellite in force-free space sweeps stationary interplanetary dust at a rate dM/dt = αv, where M is the mas

Question

A satellite in force-free space sweeps stationary interplanetary dust at a rate dM/dt = αv, where M is the mass, v is the velocity of the satellite, and α is a constant. What is the deceleration of the satellite?

Accepted Answer

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

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