Question

If a satellite collects the rate of unforeseen intergenerative dust ((dM)/(dt))=av in the free interiors, then the satellite will accelerate

Answer 1

Answer on Question#78945 - Physics - Astronomy - Astrophysics

If a satellite collects the rate of unforeseen intergenerative dust $((\mathrm{dM}) / (\mathrm{dt})) = \mathrm{av}$ in the free interiors, then the satellite will accelerate.

Solution:

Dust creates the net force $F$ given by:

$$F = - \frac {d M}{d t} v$$

According to the Newton’s second law we have

$$M \frac {d v}{d t} = F = - \frac {d M}{d t} v$$ $$M \frac {d v}{d t} + \frac {d M}{d t} v = 0$$ $$\frac {d (M v)}{d t} = 0$$ $$M v = p _ {0}$$

$p_0$ – initial momentum of the satellite (before encountering dust).

Since $\frac{dM}{dt} = av$, we obtain

$$M \frac {1}{a} \frac {d M}{d t} = p _ {0}$$ $$\frac {d (M ^ {2})}{d t} = 2 a p _ {0}$$ $$M = \sqrt {M _ {0} ^ {2} + 2 a p _ {0} t},$$

where $M_0$ – initial mass of the satellite.

Substituting this and $\frac{dM}{dt} = av$ in to equation (1) we obtain:

$$\sqrt {M _ {0} ^ {2} + 2 a p _ {0} t} \frac {d v}{d t} = - a v ^ {2}$$ $$- \frac {d v}{v ^ {2}} = a \frac {d t}{\sqrt {M _ {0} ^ {2} + 2 a p _ {0} t}}$$ $$\frac {1}{v} = \frac {1}{p _ {0}} \sqrt {M _ {0} ^ {2} + 2 a p _ {0} t}$$ $$v = \frac {1}{\frac {\sqrt {M _ {0} ^ {2} + 2 a p _ {0} t}}{p _ {0}}} = \frac {v _ {0}}{\sqrt {1 + \frac {2 a p _ {0} t}{M _ {0} ^ {2}}}}$$

Thus the acceleration:

$$\frac {d v}{d t} = - \frac {\frac {a p _ {0} v _ {0}}{M _ {0} ^ {2}}}{\left(1 + \frac {2 a p _ {0} t}{M _ {0} ^ {2}}\right) ^ {3 / 2}}$$

Answer: acceleration: $-\frac{\frac{ap_0v_0}{M_0^2}}{\left(1 + \frac{2ap_0t}{M_0^2}\right)^{3/2}}$ .

Answer provided by https://www.AssignmentExpert.com