A particle, starting from rest, moves in a straight line with constant acceleration `a_0 ` for a time interval `t_0`.Then is accelerates at a constant rate `2a_0` for time interval ` 2t_0` .The distance travelled by the particle in time ` 3t_0` from the starts is S . Then ` (S)/(a_0t_0^(2) )= `

Explanation

To solve the problem, we need to calculate the total distance \( S \) traveled by the particle during the time intervals given, and then find the ratio \( \frac{S}{a_0 t_0^2} \).

### Step-by-Step Solution:

1. **Calculate the distance traveled in the first time interval \( t_0 \)**:
- The particle starts from rest, so the initial velocity \( u = 0 \).
- The acceleration during this interval is \( a_0 \).
- Using the formula for distance:
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