Which Particle Reaches the Other End of the Hemisphere First?

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Two particles of equal mass and size are released simultaneously from opposite ends of a diameter on a frictionless hemispherical surface and along a wire. The particle on the hemisphere accelerates due to gravitational force, increasing its speed as it descends. Although the path along the hemisphere is longer, the increase in speed compensates for the distance. Ultimately, the particle traveling along the hemisphere reaches the other end first. The discussion highlights the role of gravitational force in accelerating the particle on the curved surface.
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Two particles of same mass and size are released SIMULTANEOUSLY from one end of a diameter of a hemispherical surface, along a wire joining the diameter and along the hemispherical surface respectively..The hemispherical surface is frictionless AND WIRE HAS NO TENSION..Both have a horizontal velocity v.

WHICH REACHES THE OTHER END FIRST?
 
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The speed of the particle on the hemisphere gradually rises because of the force present along the hemisphere surface.And once it has reached the bottom most point its speed decreases gradually and becomes v again by conservation of energy principle.So even though the path of the body traveling along the hemisphere is more but the speed increase more than makes up for it.And hence particle going along the hemisphere reaches first.

I don't have any mathematical explanation.Srry.
 
what force along the " hemisphere"?
 

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