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So I do not understand how, in an inelastic system, momentum is conserved.

Consider this problem:

Tarzan swings from a branch 4 meters up and picks up jane how far can he swing up the other side?

He weighs 100kg and is standing still on the branch. He jumps off and picks up 60kg Jane who is standing still on the ground. How high of a branch can he reach on the other side?

At the top, Tarzan has total energy = mgh = 3920J

When he goes down to pick up Jane, why doesn't this work:

Ek = (0.5)(160)(v^2)

v = 7m/s

Why must we get velocity using Ek formula before Tarzan picks her up, and then use conservation of momentum to find new velocity, and then use Ek to get the new total energy (where there is a loss) and then find the new height. So like this:

3920 = (0.5)(100)(v^2)

v=8.85m/s

Momentum Before pickup = Momentum after pickup

v = 5.53m/s (as opposed to 7m/s)

And the rest.

WHERE is the energy lost?! Both mathematically and conceptually. I mean, we are in a closed system and assuming no friction, right?

I am just perplexed by this whole unit.

I understand that energy is lost through sound and heat, but it should not be going on here.