Question

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?

Answer 1

Jane's mechanical energy at any time is
$E=U+K$
where $U=mgh$ is the potential energy, while $K= \frac{1}{2} mv^2$ 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:
$E=K= \frac{1}{2} mv^2$
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:
$E=U=mgh$

But E must be conserved, so the initial kinetic energy must be equal to the final potential energy:
$\frac{1}{2}mv^2=mgh$
from which we can find h, the maximum height Jane can reach:
$h= \frac{v^2}{2g}= \frac{(6.8 m/s)^2}{2\cdot 9.81 m/s^2}=2.36 m$