Answer
Tarzan's speed just before he reaches Jane is 7.89 m/s.
Work Step by Step
We can find the y-component of the rope's length when the angle is $45^{\circ}$
$y_1 = 20~cos(45^{\circ}) = 14.14~m$
We can find the y-component of the rope's length when the angle is $30^{\circ}$
$y_2 = 20~cos(30^{\circ}) = 17.32~m$
We can find the difference in the starting height and the final height.
$\Delta h = 17.32~m-14.14~m = 3.18~m$
We can use the change in height to find Tarzan's speed just before he reaches Jane.
$\frac{1}{2}mv^2 = mg~\Delta h$
$v^2 = 2g~\Delta h$
$v = \sqrt{2g~\Delta h}$
$v = \sqrt{(2)(9.80~m/s^2)(3.18~m)}$
$v = 7.89~m/s$
Tarzan's speed just before he reaches Jane is 7.89 m/s.
