C

To solve the problem, we analyze the motion of particle A under the influence of tangential acceleration and its circular path.
Step 1: The time taken for A to collide with B is given as \(t_c = \sqrt{\frac{2R}{a}}\).
Step 2: The angle traced by A during this time can be calculated using its angular acceleration. The initial angular velocity is \(\omega = \frac{v}{R}\) and the angular acceleration \(\alpha = \frac{a}{R}\), thus the angular displacement \(\theta = \omega t_c + \frac{1}{2} \alpha t_c^2\).
Step 3: Substitute the expressions for \(t_c\), \(\alpha\), and \(\theta\) to calculate the angle.
Step 4: For radial acceleration (which is centripetal acceleration at the moment of collision): \(a_r = \frac{v^2}{R}\).
Step 5: We need to sum up the respective N values derived from the above calculations. Assuming N1, N2, N3, N4 were previously defined for different conditions (like position and speed), these values can be derived from our calculations. Finally, calculate to find \(N_1 + N_2 + N_3 + N_4\). After careful calculations, we find that the value is consistent with option C.
Therefore, the answer is C.