Question

A ball of mass $m$ is thrown upward with a velocity $u$. If air
exerts an average resisting force $F$, the velocity with which the
ball returns back to the thrower is:

• A. $u\sqrt{\frac{mg}{mg+F}}$
  
• B. $u\sqrt{\frac{F}{mg+F}}$
  
• C. $u\sqrt{\frac{mg-F}{mg+F}}$
  
• D. $\sqrt{\frac{uF}{mg+F}}$
  

Answer 1

The correct option is C $u\sqrt{\frac{mg-F}{mg+F}}$

We know,
$v^2=u^2+2as$
When going up
Retardation due to air friction $a^{\prime }=F/m$
Net retardation $a=g+F/m$
$u^2=2(g+\frac{F}{m})s$
$\Rightarrow s=\frac{u^2}{2(g+\frac{F}{m})}$
When coming down
Net acceleration $a=g-F/m$
$v^2=2\times (g-\frac{F}{m})\frac{u^2}{2(g+\frac{F}{m})}$
$v^2=\frac{gm-F}{gm+F}{u}^2$
$v=u\sqrt{\frac{mg-F}{gm+F}}$