A situation to compare time taken by two objects This randomly came up in my mind.

suppose a bead like particle P at A in a frictionless hemispherical bowl. It is released from A at t = 0. A horizontal velocity v is imparted to bead P. A bead Q of the same mass as P is ejected from A at the same time 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?

Kindly neglect the distortion in shape they are perfect point particles


For a particle moving along the string
$T=2R/v$
Particle along spherical surface gave me a tough time and I feel it undergoes a circular motion so time would be half of its vertical time period but I can't calculate the time period of a vertical circle.
Is there an intuitive approach for this?
 A: The idea is that the horizontally moving particle has no extra forces on it whilst the particle rolling down the ramp has the 'normal force' accelerating it while it moves till bottom and then normal force 'deaccelerates' it as it rises again.
The reason is that while supporting its weight, the normal force also gives a horizontal component of force parallel to the surface to push the object (*). So, the thing is that even though totally it the velocity is the same at the end and starting point for both, the particle rolling down the ramp got pushed 'quicker' into the ending point.
For figuring out the time of the particle's motion it may be a bit complex,  but I myself have tried to do it and it turns out to be some complex math (see here)

*: Think of why objects move down ramps, this is also why we need frictions to keep things from sliding down ramps
On comment's by OP, I reference these other answers which discuss how 'constraint' forces accelerate object whilst no work is done
See here and here
