Question

Adolf and Ed are wearing harnesses and are hanging at rest from the ceiling by means of ropes attached to them. Face to face, they push off against one another. Adolf has a mass of 117 kg, and Ed has a mass of 76 kg. Following the push, Adolf swings upward to a height of 0.67 m above his starting point. To what height above his own starting point does Ed rise?

Answer 1

Not sure if this is right, but I will do my best. I would use gravitational potential energy.

Adolf - m=117, h=0.67 and g=9.81
GPE = mhg
GPE = 117*0.67*9.81
GPE = 769J

Now we know how much potential energy Adolf has and since they both had the same force applied to them, we can assume that Ed will have this same energy.

Ed - m=76, h=?, g=9.81 and GPE=769
GPE = mhg
769 = 76*9.81*h
h = 1.031m

Therefore Ed goes 1.031m above his starting position

Answer 2

Let's consider Adolf and Ed collectively our physical system. There is no EXTERNAL force acting on the system (Of course, there is gravitational force but that balanced by the tension in the ropes.). The pushing off forces are INTERNAL ones and we do not know anything about these forces. When we do not know about forces, conservation laws are very handy to understand the behavior of the system.

1) Conservation of linear momentum
2) Conservation of energy

The system is initially at rest, so linear momentum and kinetic energy of this system are zero.
m1 = 117 kg, m2 = 76 kg, h1 = 0.67 m ,H = ??
Just after pushing off the kinetic energy of Adolf = m1v1^2 /2 which becomes potential energy at a height of h1 = m1*g*0.67
So \[v _{1} = \sqrt{2gh_{1}}\]


Linear momentum after pushing off = Linear momentum before pushing off
m1v1-m2v2 = 0 (Since v1 and v2 are in opposite directions !)
So, v2 = m1v1/m2

Just after pushing off the kinetic energy of Ed = m2v2^2 /2 which becomes potential energy at a height of H = m2*g*H
So \[H = \frac{ v _{2} ^2}{2g }\]
Putting v2 = m1v1/m2
\[H = \frac{ m _{1}^2 v _{1}^2}{ 2gm _{2}^2}\]
Putting \[v _{1} = \sqrt{2gh_{1}}\] we get
\[H = \frac{ m _{1}^2 h_{1}}{m _{2}^2}\]
Putting the values of m1,m2 and h1 we get
H = 1.588 m