Solve Min Velocity for Ball Not to Touch Hemispherical Rock

[012anonymousx]

Homework Statement

A person standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it a horizontal velocity v.
What is the minimum initial speed to ensure the ball doesn't touch the rock?

Homework Equations

x^2 + y^2 = r^2
y = -0.5gt^2 + R

The Attempt at a Solution

R - (gx^2)/(2v^2) > sqrt(R^2 - x^2).
The left side is eqn for parabolic trajectory, the right is the boulder

After a lot of math you get something like this:

(g^2x^4) / (4v^4) + gRx^2 / v^2 + x^2 > 0

Now I am super confused about this part:
For some reason, the claim goes like, as x approaches 0, we get the tightest limit, therefore it needs the largest curvature at the start and it will pass the boulder (reasonable I guess..).
THEN for some reason, 1 > gR / v^2

I have no idea where this came from.

See here; http://minerva.union.edu/labrakes/2_D_Motion_Problems_Solutions.pdf

Another solution I read was;

m * v^2/R > mg
Fc > Fg.

Why...? The acceleration into the boulder has to be GREATER than gravity?
That doesn't make a lot of sense to me :(

 

[NascentOxygen]

A person standing at the top of a hemispherical rock of radius R kicks a ball (initially at rest on the top of the rock) to give it a horizontal velocity v.
What is the minimum initial speed to ensure the ball doesn't touch the rock?

At first glance, ISTMT you need the ball to travel at least R metres horizontally in the time that it takes to fall R metres vertically.