Question

A solid metallic sphere of radius $r$ is allowed to fall freely through air. If the frictional resistance due to air is proportional to the surface area and to the square of the velocity, then the terminal velocity of the sphere is proportional to which of the following ?

• A. $r^2$
• B. $r$
• C. $r^{3/2}$
• D. $r^{1/2}$

Answer 1

The correct option is D $r^{1/2}$

Given that resistive force
$f\propto A{V}^2$
$f=KA{V}^2$
When sphere attains terminal velocity, it is in equilibrium
$mg=KA{V}^2$
$\frac{4}{3}\pi r^3\rho g=K\left(4\pi r^2\right)V^2$
$\therefore V\propto \sqrt{r}$