91Ó°ÊÓ

i) Mass of man on the ladder, m is 80 kg .

ii) Mass of balloon with the person, M is 320 kg .

iii) Speed of man on ladder relative to the ground, $v_{m}is2.5m/s$.

You use the concept of center of mass. When the man climbs on the ladder, the center of mass of man – the balloon system will not move. You can find the speed of the balloon by using the equation of center of mass of velocity. The formula required is given below.

$V_{cm}=\frac{m{v}_m-M{v}_b}{m+M}$

As the man on theladder is moving upward, the center of mass of the man-balloon system will not move.

Thus, the direction of motion of the balloon will be downward.

We can use the equation of center of mass for velocity,

$V_{cm}=\frac{m{v}_m-M{v}_b}{m+M}$

Where,$v_m$is the speed of the man relative to the ground and$v_b$is the speed of the balloon, and v is the speed of a man relative to the ladder.

$v_m=v-v_b$

Plugging the values, we get,

$$\begin{gathered} V_{cm}=\frac{m\left(v-v_b\right)-M{v}_b}{m+M} \\ \frac{mv-\left(m+M\right)v_b}{m+M}=0 \\ {v}_b=\frac{mv}{m+M} \end{gathered}$$

Substitute the values in the above expression, and we get,

$$\begin{gathered} v_b=\frac{\left(80×2.5\right)}{\left(80+320\right)} \\ {v}_b=\frac{200}{400} \\ =0.5m/s \end{gathered}$$

Thus, the speed of the balloon, $v_{b}is0.5m/s$.

When the man stops climbing, there is no relative motion between the man on the ladder and the balloon.

Thus, the speed of the balloon will be zero.