Pendulum problem where it hits a peg swings around and lands on top of it Using energy equation from when it is released to when it reaches its highest point after hitting the peg I get change in kinetic energy = 0 and change in potential due to gravity = 0
Does the string go slack at this point?
The height at this point would be (L-Lcos(theta)).
I'm trying to find some sort of relationship so I can get the x component of the velocity after it turns into a projectile

 A: The way I would start the problem is by dividing it up in subproblems:


*

*Express the speed $v(\theta)$ of the ball at angle $\alpha$ 

*Calculate the angle $\beta$ (between the rope at angle $\alpha$ and the part of the rope with the ball attached) that the ball will travel in circular motion upwards until the tension drops to $0$ and the motion becomes projectile motion. Together with the angle, you should also have calculated the velocity $v'(\theta)$ at that point. 

*Now we have available: the launch coordinates of a projectile motion (can be calculated by using $\beta$), the launch velocity of the projectile motion (by using $v'(\theta)$) and the point where it needs to land (set those coordinates to $(0,0)$)


*Use the constraint that the ball has to land on the peg to remove the final unknowns in the problem.



In detail:


*

*Try using the conservation of energy to get the speed of the ball in function of the starting height and express the height as a function of $\theta$

*The tension in the rope becomes $0$ when the centripetal acceleration is completely dealt with by gravity. In that case the rope doesn't have to exert any force on the ball. To find the angle at which this event occurs you could project the gravitational acceleration on the smaller part of the rope and set it equal to the centripetal acceleration required for the circular motion of the ball. This results in a relation between $\beta$ and $v'(\theta)$. 
Another relationship between $\beta$ and $\theta$ can be found by using the conservation of energy again, expressing the energy of the ball at the top of the second circular trajectory in function of the energy at the bottom of the trajectory. This will give jet another relation between $\beta$, $v$ and $v'$ 

*At this moment there are 4 unknowns in the game: $\theta$ (which we will need to express as a function of the initial parameters),$v$, $\beta$ and $v'(\theta)$ and 3 equations: one linking $v$ and $\theta$, one linking $v$, $v'$ and $\beta$ and one linking $\beta$ and $v'$. 
By using the second equation you can eliminate $v$ and end up with 2 equations and 3 unknowns. 

*The final part of the exercise consists of requiring the ball to land on the peg. By using the (unknown) $\beta$ it is possible to find the coordinates of the ball at the point of $0$ tension. Then by using $v'$ this reduces to a projectile motion problem. Set up the equations required for the ball to hit the peg and you will find yet another constraint on $\theta$, $\beta$ and $v'$.
3 unknowns and 3 equations: now you should be able to eliminate $\theta$ from these equations. 
I deliberately did not write all the details on how to solve the problem but if you happen to get stuck somewhere, I'll be happy to help you out. 
