- #1
stfz
- 35
- 0
Homework Statement
A 52kg man is on a ladder hanging from a balloon that has a total mass of 450kg (including the basket passenger). The balloon is initially stationary relative to the ground. If the man on the ladder begins to climb at 1.2m/s relative to the ladder, (a) in what direction does the balloon move? (b) At what speed (with respect to the ground) does the balloon move? (c) If the man stops climbing, what is the speed of the balloon?
Homework Equations
Centre of mass velocity: ##v_{CM}=\frac{1}{M}\sum m_i v_i##
Galilean transformation: ##v'=v-v_{frame}##
The Attempt at a Solution
My question is with part (b). I know that, relative to the balloon, the man is moving up at 1.2m/s. However, the balloon is also moving down so I must find the man's velocity relative to the ground (and hence the desired absolute velocity of balloon). I've been able to compute the correct answer using a different method (##v_{balloon}=-0.124m/s##).
Nevertheless, I'm not sure why the following method is not working - I have a misconception somewhere.
I'm going to assume that momentum is conserved (i.e. the balloon and man are in equilibrium, with the air supporting them against gravitational forces). The system is also initially stationary, hence total momentum cannot change.
Thus, ##mv_m+Mv_b=0##, where ##v_m, v_b## are absolute velocities of man and balloon respectively, with masses ##m, M## respectively.
I have 2 unknowns, but I know that, in balloon frame, ##v_m'=1.2##. So I transform to balloon frame by subtracting ##v_b##:
##v_b'=0## and ##v_m'=v_m-v_b##.
In this new frame of reference, I know that momentum is still conserved. Also, initially, when both balloon and man are stationary, the momentum in this frame is zero.
Hence, ##mv_m'+Mv_b'=mv_m'=m(v_m-v_b)=0##.
This seems to imply ##v_m=v_b## which is unrealistic and incorrect.
I've made a mistake somewhere in my reasoning above, in particular when I began to transform. Could someone enlighten me as to what I've misunderstood?
The method I used (which worked) was to recognize that ##v_m'=v_m-v_b=1.2## and hence I have a linear relationship between v_m and v_b. Solving along with the momentum conservation condition yields ##v_b=-0.124##.
Thanks!
Stephen