Solve Ball on a String for Velocity w/ Mass, Length & Gravity

[melancholy2]

This isn't actually a homework problem, but I thought it would be relevant to post it here since it is a problem afterall, and I need some help at the solution. Thank you everyone!

Homework Statement

There is a ball mass M hanging on a massless string length L, from a nail in the wall. When the ball is projected with initial horizontal velocity V, it swings up, but then falls back down and hits the nail. Find V in terms of M L and g.

2. Requested Help

I would appreciate it if someone could see if my method is problematic, and if my answer is correct if you would want to work it out.

The Attempt at a Solution

For ball to leave a circle and transit to a parabolic motion, Tension in string must equal to zero at point of transition. Thereafter, the parabolic motion will coincide with the nail. I have 3 equations and 3 unknowns - velocity at moment of transition, time taken for parabolic flight, and angle from vertical which transition occurs.

Using the above, I got an angle of 65.53 deg
I got v^2 = gl cos(angle)
And I eliminated t.

Finally, after using conservation of energy, I arrive at projection velocity V = Sqrt(3.24Lg)

 

[tiny-tim]

Welcome to PF!

Hi melancholy2! Welcome to PF!

(have a square-root: √ and a degree: º and try using the X² tag just above the Reply box )

For ball to leave a circle and transit to a parabolic motion, Tension in string must equal to zero at point of transition. …

Yes, that's correct. But it's very difficult to check your final result without seeing your full calculations.

 

[melancholy2]

Hi everyone,

I've relooked at my workings and managed to come up with a new answer which I think is correct. Angle is now 54.7 degrees. Please see the attachment solution. Thanks!

 

[tiny-tim]

Hi melancholy2!

(I haven't actually checked the last 6 lines , but apart from that …)

Yes, that looks fine.

(except you could have saved yourself a little trouble if you'd noticed that lcosθ + lsin²θ/cosθ = l/cosθ )