Question

A balloon rises from ground with an initial velocity of zero, and its acceleration decreases with height linearly. Its acceleration at ground is $a_0$ and decreases to zero at height $H_0$ then

• A. Maximum velocity of balloon will be $\sqrt{a_0{H}_0}$
• B. Maximum velocity of balloon will be $2\sqrt{a_0{H}_0}$
• C. Velocity when it is at height $\frac{H_0}{2}$ is $\frac{\sqrt{3a_0{H}_0}}{2}$
• D. Velocity when it is at height $\frac{H_0}{2}$ is $\sqrt{a_0{H}_0}$

Answer 1

Answer: (A), (C)

Velocity when it is at height $\frac{H_0}{2}$ is $\frac{\sqrt{3a_0{H}_0}}{2}$

$a=a_0-\frac{a_0}{H_0}h$
Using $vdv=adh$,
$\Rightarrow {\int }_0^{v}vdv={\int }_0^h(a_0-\frac{a_0}{H_0}h)dh$
$v=\sqrt{2a_{0}h-\frac{a_0}{H_0}{h}^2}$
v is maximim at, $h=H_0$
at $h=H_0,v=\sqrt{a_0{H}_0}$
at $h=\frac{H_0}{2}\Rightarrow v=\frac{\sqrt{3a_0{H}_0}}{2}$