Question

Jane is looking for Tarzan is runnning at top speed of 5.3 m/s. She grabs a vine hanging vertically from a tall tree. How high can she swing

Answer 1

According to the classical dynamics, if we assume any loss of mechanical energy is not considered here. Then the total mechanical energy is conserved and then for a system the initial total mechanical energy is equal to the final mechanical energy. Therefore if we consider that when jane is on the ground moving with a speed of 5.3m/s then Jane has only kinetic energy as total mechanical energy, Thus.

where M.E. stands for total mechanical energy, K.E. is for kinetic energy and P.E. for potential energy, suffix i and f are for initial and final study.
also we assume m as jane's body mass and v as jane's velocity on ground.

Now when jane is hanging a vine to reach the highest point and the total mechanical energy is converted from kinetic energy to potential energy completely. Thus,

Here we consider the maximum highest point of Jane is h.

Due to the conservation of total mechanical energy  



That means the maximum height of Jane will be 1.43 m.