Question

Genius please help me with the qs

Consider an expanding sphere of instantaneous radius R whose total
mass remains constant. The expansion is such that the instantaneous
density d remains uniform throughout the volume. The rate of fractional
change in density is constant. The velocity v of any point on
the surface of the expanding sphere is proportional to_____​

Answer 1

Answer:

v ∝ $R^{-2}$

Explanation:

Mass = Density x Volume

$M = \frac{4}{3}\pi R^{3} d$

Differenciating both sides with t

$\frac{dM}{dt} = \frac{4}{3} \pi \frac{d(R^{3} D)}{dt}$

dM/dt is 0 since mass is constant

$\frac{d(R^{3}D )}{dt} = 0\\3R^{2} \frac{dR}{dt} + \frac{dD}{dt} = 0$

dR/dt = v

dD/dt is constant, so v is proportional to $R^{-2}$