Particle on top of a half circle

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To determine the minimum initial velocity required for a point-particle to clear a half-circle of radius R, the particle must be analyzed under the influence of gravity. The particle starts at rest at the top of the half-circle, and the challenge lies in ensuring it does not roll down either side. The initial velocity must be sufficient to allow the particle to travel a distance greater than R before descending. A consideration of projectile motion and the parabolic trajectory is essential in calculating this velocity. The discussion emphasizes the need for a mathematical approach to relate the initial velocity to the radius and gravitational acceleration.
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Homework Statement



A point-particle sits at rest at the top of a half-circle with radius R. Find the minimum initial velocity v0 the particle has to have in order to clear the half circle without rolling down on one of its sides.

Homework Equations



None are given.

The Attempt at a Solution



Since the radius is given as a variable, I'm guessing it should be possible to write the solution as a function of R and g. The only thing I've managed to write down is:

v0x = ?
v0y = 0
ax=0
ay=g

Any help on how to proceed? I have no idea how to make sure the particle clears the circle... I thought about solving for the initial velocity needed to make sure the particle travels farther than R before hitting the ground, but the path will be a parabola and not a circular path, so again I'm not sure. Thanks!
 
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... I thought about solving for the initial velocity needed to make sure the particle travels farther than R before hitting the ground...

I think you are on the right track.
 

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