Question

A small spherical ball of radius r, falling through a viscous medium of negligible density has terminal velocity 'v'. Another ball of the same mass but of radius 2r, falling through the same viscous medium will have terminal velocity:

• A. $\frac{v}{2}$
• B. $\frac{v}{4}$
• C. $4v$
• D. $2v$

Answer 1

Answer: (A)

$\frac{v}{2}$

Answer 2

Since the density of the medium is negligible, the buoyancy force can be ignored. At terminal velocity, the gravitational force on the ball is balanced by the viscous drag force. The terminal velocity $v$ is given by:

$v \propto \frac{1}{r},$

for a sphere of constant mass.

Let the terminal velocity of the original ball (radius $r$) be $v$ and the terminal velocity of the larger ball (radius $2r$) be $v'$.

Using the inverse proportionality:

$\frac{v}{v'} = \frac{r'}{r}.$

Since $r' = 2r$:

$\frac{v}{v'} = 2 \implies v' = \frac{v}{2}.$

Thus, the terminal velocity of the larger ball is:

$\frac{v}{2}.$