Another Mechanics Question...Help me!?

Question

A small wooden cube is placed on a top of a hemisphere of radius r. What maximum horizontal velocity is to be given to the cube to detach it from the start of its motion?
(A)v^2>rg (B)v^2

Answer 1

If I understand the question, you are giving a velocity to the cube and don't want it to touch the hemisphere at any time in its flight. I think you want the minimum, not the maximum hor. velocity to do this, right? Because any arbitrarily high velocity will do it. I'll use a planar analysis where the hemisphere is a semicircle.
Assuming the semicircle is centered at x=y=0, the trajectory will be a parabola y=r-kx^2 and at y=0, |x| must be >r. Another constraint arises when we consider the region of the semicircle near the top. Here the semicircle approximates a parabola, since 1-cos(small angle) approaches small angle^2/2. Thus deltay = r(1-cos(small angle)) = r-y approaches r*small angle^2/2 = r(x/r)^2/2, or x approaches sqrt(2/r*deltay*r^2) = sqrt(2r*deltay).
Downward acceleration is g and horizontal velocity is v. When y=0, 0.5gt^2=r, so t=sqrt(2r/g), and x=vt must be >r. Thus the 1st condition (for clearance at the bottom) is v > r/sqrt(2r/g), or
v > sqrt(rg/2).
The 2nd condition requires that when r-y=deltay is small, then and x=vt must be >sqrt(2r*deltay) where t=sqrt(2deltay/g), or
v > sqrt(2r*deltay) / sqrt(2deltay/g) or
v > sqrt(rg).
This latter is the more stringent of the two requirements and thus is the minimum clearance velocity.