A particle P is sliding down a frictionless hemispherical bowl. It passes the point A at t = 0. At this instant of time, the horizontal component of its velocity is v. A bead Q of the same mass as P is ejected from A at t=0 along the horizontal string AB, with the speed v. Friction between the bead and the string may be neglected.Which bead reaches point B earlier?
The Attempt at a Solution
Let θ be the angle which the particle P makes with the vertical .
N-Mgcosθ = Mv2/R
Mgsinθ = -Mvdv/dx
or,Mgsinθ = -Mvdv/Rdθ
RMgsinθdθ = -Mvdv
∫β0RMgsinθdθ = -∫uwMvdv , where β is the initial angle ; u is the initial velocity and w is the velocity at the bottommost point .
But how to relate initial velocity 'u' with the initial horizontal component of velocity 'v' .
I am not sure if this is the correct approach.
I would be grateful if someone could help me with the problem.