A particle is moving on a circle of radius `R` such that at every instant the tangential and radial accelerations are equal in magnitude. If the velocity of the particle be `v_(0)` at `t=0`, the time for the completion of the half of the first revolution will be

Options

A. `R//v_(0)`
B. `(R//v_(0))(1-e^(-x))`
C. `(R//v_(0))e^(-x)`
D. `(R//v_(0))(1-e^(-2x))`

Explanation

To solve the problem, we need to analyze the motion of a particle moving in a circular path with equal tangential and radial accelerations. Let's break down the solution step by step.

### Step 1: Understanding the relationship between tangential and radial acceleration
Given that the tangential acceleration \( a_t \) is equal to the radial (centripetal) acceleration \( a_r \), we can express these accelerations as:
- Tangential acceleration: \( a_t = \alpha R \) (where \( \alpha \) is the angular acceleration)
- Radial acceleration: \( a_r = \frac{v^2}{R} \)

Since \( a_t = a_r \), we have:
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