Answer
The correct answer is:
(a) A has the larger terminal velocity.
Work Step by Step
We can write the expression for the magnitude of the drag force for a sphere falling through a viscous fluid:
$F_d = 6\pi \eta ~r~v$
A falling object reaches terminal velocity when the opposing drag forces are equal in magnitude to the gravitational force:
$mg = F_d$
$mg = 6\pi \eta ~r~v$
$v = \frac{mg}{6\pi \eta ~r}$
We can use the density $\rho$ to find an expression for the mass of a sphere with radius $r$:
$m = (\rho)(\frac{4}{3}\pi~r^3)$
We can replace this expression for the mass $m$ in the terminal velocity equation:
$v = \frac{mg}{6\pi \eta ~r}$
$v = \frac{(\frac{4}{3}\rho \pi~r^3)g}{6\pi \eta ~r}$
$v = \frac{2~\rho \pi~r^2~g}{9\pi \eta}$
From this equation, we can see that for spheres with the same density $\rho$, the sphere with a larger radius will have a larger terminal velocity. Since A has a larger radius, A has a larger terminal velocity.
The correct answer is:
(a) A has the larger terminal velocity.
