Physics
posted by Becky .
Tarzan tries to cross a river by swinging from one bank to the other on a vine that is 10.2 m long. His speed at the bottom of the swing is 7.6 m/s. Tarzan does not know that the vine has a breaking strength of 1.0 103 N. What is the largest mass that Tarzan can have and still make it safely across the river?

At the bottom, Tarzan has a speed of
v = 7.6 m/s
This means that at that point he is actually accelerating upoward at the centripetal acceleration of:
a = v^2/r
where
r = 10.2 m
The centripetal acceleration arises because if you change direction the velocity vector changes, even if the speed itself doesn't change.
You then apply Newton's second law:
F = m a
The total force acting on Tarzan must be equal to his mass times his acceleration. If you take the upward direction as positive, then you can write this as:
F_vine  m g = m a
Where F_vine is the force exerted on Tarzan by the vine and m g is, of course the force exerted by the Earth's gravity field on Tarzan, which enters the equation with a minus sign because we've chosen the convention to take the upward direction as positive.
So, you see that:
F_vine = m (a + g)
Then, by Newton's third law, the force exerted by the Vine on Tarzan is minus the force exerted by Tarzan on the vine. Now the magnitude of this force can be 1.0 10^3 N at most. So, you can use this to solve for the maximum value for m. 
thank you sooo much!