How Does a Balloon React When a Man Climbs a Ladder Attached to It?

[exparrot]

Homework Statement

In the figure below, a 74 kg man is on a ladder hanging from a balloon that has a total mass of 260 kg (including the basket passenger). The balloon is initially stationary relative to the ground. The man on the ladder begins to climb at 2.5 m/s relative to the ladder.

(a) In what direction does the balloon move? My answer is downwards which is correct

(b) At what speed does the balloon move?

Homework Equations

v_mg = v_mb - v_bg

(where the subscripts _mg refer to man relative to ground, _mb man relative to balloon (ladder), and _bg balloon relative to ground)

v_com = [m_mg - Mv_bg]/M + m

The Attempt at a Solution

I actually have the worked out solution to this problem but I'm finding it hard to grasp that
v_com = [m_mg - Mv_bg]/M + m = 0 (as per my textbook solutions). This is my understanding: the balloon is stationary relative to the ground, thus the v_com of the balloon-ground system would be 0. However, the man starts to move up the ladder causing the balloon to move downward, thus changing the v_com. How can, then, the textbook solution in solving for v_bg set the aforementioned equation for v_com to 0? I mean, the v_bgto be calculated is not when the balloon is stationary relative to the ground, obviously, it's changing relative to the ground (moving downwards) while the man is moving up the ladder. I would appreciate help in understanding this conceptually, especially if my thinking is wrong.

Thanks!

 

[rl.bhat]

If xp, xb and xm are the distances of passenger, balloon and the center of mass from the ground, then
xm = (m*xp + M*xb)/(M + m)
If you take the derivative with respect to time to find their velocities, we get
v(xm) = (m*vp + M*vb)/(M + m). Since vp and vb are in the opposite direction the final expression becomes
vm = ...?

 

[exparrot]

If xp, xb and xm are the distances of passenger, balloon and the center of mass from the ground, then
xm = (m*xp + M*xb)/(M + m)
If you take the derivative with respect to time to find their velocities, we get
v*xm = (m*vp + M*vb)/(M + m). Since vp and vb are in the opposite direction the final expression becomes
vm = ...?

I'm sorry, I don't understand. Why should I have to take the derivative if I already have an equation tailored to this situation? Maybe I'm not understanding something...

 

[rl.bhat]

In the problem there are three center of masses.
1) Center of mass of man
2) Center of mass of balloon
3) Center of mass of ( man + balloon) system.
Since there is no external force acting on the system, center of mass of the system remains at rest. So v(cm) is zero.