Answer
Vine doesn't break
Work Step by Step
Lets calculate the speed at the end of the descent ( initial speed is zero so kinetic energy will be zero too). The total energy of tarzan initially is:
$E_{tot}=E_{k0}+E_{p0}=0+mgh(1)$
$E_{k0} $ and $E_{p0}$ are initial kinetic and potential energies of tarzan. After the descent, the potential energy will be zero and all potential energy will be converted to kinetic energy. Therefore,
$ E_{tot}=E_{k_{1}}+E_{p1}=\dfrac {mv^{2}_{1}}{2}+0\left( 2\right)$.
So from (1) and (2), we get:
$E_{tot}=E_{k_{1}}+E_{po}=\dfrac {mv^{2}_{1}}{2}+0=mgh+0\Rightarrow v^{2}_{1}=2gh(3)$
Lets assume the vine doesn't break until tarzan reaches the lowest point. The force applied to vine by tarzan at the lowest point (trajectory of tarzan will be circular) will be:
$F_{tot}=mg+\dfrac {mv^{2}_{1}}{L}\left( 4\right) $ where $L$ is the length of the vine
Lets calculate this using (3). We get:
$F_{tot}=mg+\dfrac {mv^{2}_{1}}{L}=mg+\dfrac {2mgh}{L}=mg\left( 1+\dfrac {2h}{L}\right) \approx 932.62N < F_{vine}=950N$
$F_{vine} $ is the minimum force to break the vine so vine doesn't break.
