Tarzan swings on a vine from a cliff to rescue Jane, who is standing on the grou

Question

Tarzan swings on a vine from a cliff to rescue Jane, who is standing on the ground surrounded by snakes. His plan is to push off the cliff, grab Jane at the lowest point of his swing, and carry them both to the safety of a nearby tree (see the figure). Tarzan's mass is 80 kg, Jane's mass is 42 kg, the height of the lowest limb of the target tree is 12 m, and Tarzan is initially standing on a cliff of height 15 m. The length of the vine is 30 m. With what speed should Tarzan push off the cliff if he and Jane are to make it to the tree limb successfully?

Explanation / Answer

Final Energy Conservation
1/2*(mt+mj)*v2^2 = (mt+mj)*g*h

Solve for v2:
v2 = sqrt( 2*g*h )
v2 = sqrt(2*9.8*12)
v2 = 15.33 m/s

Now we solve for Tarzan's velocity just before he grabs Jane:
Conservation of Momentum
mt*v1 + mj*0 = (mt + mj)*v2

Solve for v1:
v1 = v2*(mt+mj)/mt
v1 = v2*(80+42)/80
v1 = 1.525*v2
v1 = 23.37 m/s

Finally solve for initial velocity of Tarzan jumping from cliff.

Initial Energy Conservation
mt*g*H + 1/2*mt*v0^2 = 1/2*mt*v1^2

Solve for v0:
v0 = sqrt( v1^2 - 2*g*H )

= sqrt(23.37^2 - 2*9.8*15)
= sqrt(546.15 - 294)
= sqrt(252.15)
= 15.87 m/s

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