# Return of the Particle Race

**Classical Mechanics**Level 5

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