Jane, looking for Tarzan, is running at top speed (6.2 m/s) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward?

Answer 1

You just need to calculate Jane's kinectic energy before she grabs the vine (.5mv^2). This will be equal to her gravitational potential energy (mg*height) at the top of her swing.
.5mv^2=mgh
-cancel the mass on both sides
.5v^2=gh
19.22=9.8h
h=1.96

Answer 2

@Peter What about the non-conservative work lost with the tension force?