Question

A solid rubber ball of density 'd' and radius 'R' falls vertically through air. Assume that the air resistance acting on the ball is F $=$ KRV where K is constant and V is its velocity. Because of this air resistance the ball attains a constant velocity called terminal velocity $V_T$ after some time.Then $V_T$ is:

• A. $\frac{4\pi R^{2}dg}{3K}$
• B. $\frac{3K}{4\pi R^{2}dg}$
• C. $\frac{4}{3}\frac{\pi r^{3}dg}{K}$
• D. $\pi rdgk$

Answer 1

The correct option is A $\frac{4\pi R^{2}dg}{3K}$

Let any time t, the velocity be V.
Acceleration is then $\frac{mg-KRV}{m}=\frac{dV}{dt}$
but once it attains terminal velocity $V_T$; $\frac{dV}{dt}=0$
$\Rightarrow mg=KR{V}_T$
$\left(\frac{4}{3}\pi R^3\right)dg=KR{V}_T$
$\Rightarrow V_T=\frac{4\pi R^{2}dg}{3K}$