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 cross-sectional 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 $r^{1/2}$

The frictional resistance $f_r$ is given as:
$f_r\propto A$ and $f_r\propto v^2$
$f_r=kA{v}^2$(here k is a constant)

The downwards force is given as: $mg=\frac{4}{3}\pi r^3\rho g$

Balancing the above we get the terminal velocity as:
$\frac{4}{3}\pi r^3\rho g=k\left(\pi r^2\right)v^2$
$r\propto v^2$
$v\propto r^{1/2}$