Return of the Particle Race

The old rivals, particles $$P$$ and $$Q$$ are back, yet again for another thrilling race. This time the arena is a friction-less circular track with radius $$R$$ and diameter $$AB$$.The race begins from $$A$$ and ends at $$B$$.

$$\text{ Motion of particle P}$$: It is released from point $$A$$ at an angle $$\theta$$ with the horizontal , with a velocity of $$v_{0}$$, such that it lands exactly at point $$B$$.

$$\text{ Motion of particle Q}$$: It starts from $$A$$ and moves around the circumference, such that at every instant the $$\text{tangential}$$ and $$\text{radial}$$ accelerations are equal in magnitude. The velocity of $$Q$$ at $$t=0$$ is also $$v_{0}$$.

Find the difference in time taken for particles $$P$$ and $$Q$$ (consider equal mass and dimensions), to reach point $$B$$

Details and Assumptions:

$$\bullet$$ $$R=16m$$ , $$v_{0}=20ms^{-1}$$ , $$g=10ms^{-2}$$

$$\bullet$$ Neglect air-resistance

$$\bullet$$ If you think $$Q$$ wins , add $$5.31$$ to your difference, otherwise add $$10.73$$.

$$\bullet$$ Definitely try the first part

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