A particle is moving in a circle of radius R with a constant speed v. Its average acceleration over the time when it moves over half the circle is :

Average acceleration is defined as the total change in velocity divided by the total time taken. For a particle moving in a circle of radius R with a constant speed v:

• Time Interval (t): To cover half the circumference of the circle (distance = πR), the time taken is t = πR / v.
• Change in Velocity (Δv): Velocity is a vector quantity. If the initial velocity at the start of the half-circle is v, then after moving half-way around the circle, the particle moves in the exactly opposite direction with velocity -v. The magnitude of the change in velocity is |v - (-v)| = 2v.
• Average Acceleration: It is calculated as Δv / t = 2v / (πR / v) = 2v² / πR.

In the given options, T represents the mathematical constant π. Therefore, the average acceleration is 2v² / TR, making Option C correct.