A particle starting from rest, move with uniform acceleratin. It covers a distance `x_(1)` during third second and `x_(2)` in the fifth second. Then the ratio `x_(1)//x_(2)` is

Options

A. `3//5`
B. `5//9`
C. `9//25`
D. `25//81`

Explanation

To solve the problem, we need to find the distances \( x_1 \) and \( x_2 \) covered by a particle moving with uniform acceleration during the third and fifth seconds, respectively. We will then calculate the ratio \( \frac{x_1}{x_2} \).

### Step-by-Step Solution:

1. **Understanding the Formula**:
The distance covered by a particle in the \( n \)-th second of motion under uniform acceleration is given by the formula:
\[ x_n = u + \frac{a}{2} (2n - 1) \]
where \( u \) is the initial velocity, \( a \) is the acceleration, and \( n \) is the second in which the distance is being calculated.

2. **Setting Initial Conditions**:
Since the particle starts from rest, we have:
\[ u = 0 \]

3. **Calculating \( x_1 \) (Distance in the 3rd second)**:
For \( n = 3 \):
\[ x_1 = 0 + \frac{a}{2} (2 \cdot 3 - 1) = \frac{a}{2} (6 - 1) = \frac{a}{2} \cdot 5 = \frac{5a}{2} \]

4. **Calculating \( x_2 \) (Distance in the 5th second)**:
For \( n = 5 \):
\[ x_2 = 0 + \frac{a}{2} (2 \cdot 5 - 1) = \frac{a}{2} (10 - 1) = \frac{a}{2} \cdot 9 = \frac{9a}{2} \]

5. **Finding the Ratio \( \frac{x_1}{x_2} \)**:
Now we can compute the ratio of the distances:
\[ \frac{x_1}{x_2} = \frac{\frac{5a}{2}}{\frac{9a}{2}} = \frac{5a}{2} \cdot \frac{2}{9a} = \frac{5}{9} \]

### Final Answer:
The ratio \( \frac{x_1}{x_2} \) is \( \frac{5}{9} \).

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