How High Can Tarzan Swing After Jane Lets Go?

[Pires]

Tarzan and Jane, whose total mass is 140.0 kg, start their swing on a 6.0 m long vine when the vine is at an angle of 30.0° with the horizontal. At the bottom of the arc, Jane, whose mass is 50.0 kg, releases the vine. What is the maximum height at which Tarzan can land on a branch after his swing continues?

I have tried to get this problem but i just seem to be pausing because i can't find the right information/formula to proceed.
All help appreciated

 

[Doc Al]

Energy is conserved. The energy of "Tarzan + Jane" at the starting point equals the energy they have at the bottom of the arc. When Jane let's go, Tarzan is left alone. But energy is still conserved. (You may find the problem easier than it looks. Try to do things with symbols; don't rush to plug in numbers.)

 

[quicknote]

I would approach this problem using the principles of work and energy. The first step would be to calculate the initial potential energy of Tarzan and Jane at the start of their swing, which would be given by the formula PE = mgh, where m is the total mass (140 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height (in meters).

Next, I would consider the conservation of energy. At the bottom of the arc, Jane releases the vine, causing Tarzan to continue his swing. This means that the initial potential energy is converted into kinetic energy and is equal to the kinetic energy at the bottom of the arc. The formula for kinetic energy is KE = 1/2mv^2, where m is the mass (140 kg) and v is the velocity (in m/s).

Since we know the length of the vine (6.0 m) and the angle at which it is released (30.0°), we can use trigonometry to calculate the horizontal and vertical components of Tarzan's velocity at the bottom of the arc. The horizontal component will remain constant, while the vertical component will decrease due to the force of gravity.

Using the formula v = √(v₁² + v₂²), where v₁ is the horizontal component and v₂ is the vertical component, we can calculate the total velocity at the bottom of the arc.

Now, we can use the formula KE = 1/2mv² to calculate the kinetic energy at the bottom of the arc. This value will also be equal to the potential energy at the maximum height that Tarzan can reach. We can then use the formula PE = mgh to solve for h, the maximum height.

In summary, the key steps to solving this problem would be to calculate the initial potential energy, consider the conservation of energy, use trigonometry to calculate Tarzan's velocity at the bottom of the arc, and finally use the equations for kinetic and potential energy to solve for the maximum height. I hope this helps in finding the right approach to this problem.