Jane, looking for tarzan, is running at top speed (6.8 m/s) and grabs a vine hanging vertically from a tall tree in the jungle. how high can she swing upward?

Jane's mechanical energy at any time is

where  is the potential energy, while  is the kinetic energy.

Initially, Jane is on the ground, so the altitude is h=0 and the potential energy is zero: U=0. She's running with speed v, so she has kinetic energy only:

Then she grabs the vine, and when she reaches the maximum height h, her speed is zero: v=0, and so the kinetic energy becomes zero: K=0. So now her mechanical energy is just potential energy:


But E must be conserved, so the initial kinetic energy must be equal to the final potential energy:

from which we can find h, the maximum height Jane can reach: