A body `A` starts from rest with an acceleration `a_1`. After `2` seconds, another body `B` starts from rest with an acceleration `a_2`. If they travel equal distances in the `5^(th)` second, after the start of `A`, then the ratio `a_1 : a_2` is equal to :

Options

A. `5 : 9`
B. `5 : 7`
C. `9 : 5`
D. `9 : 7`

Explanation

To solve the problem, we need to find the ratio of the accelerations \( a_1 \) and \( a_2 \) of two bodies \( A \) and \( B \) that travel equal distances in the 5th second after \( A \) starts moving.

### Step-by-step Solution:

1. **Understanding the Motion of Body A**:
- Body \( A \) starts from rest with an acceleration \( a_1 \).
- The distance traveled by body \( A \) in the 5th second can be calculated using the formula for distance in the nth second:
\[ S_n = u + \frac{1}{2} a (2n - 1) \]
- Since \( A \) starts from rest, \( u = 0 \) and for \( n = 5 \):
\[ S_5 = 0 + \frac{1}{2} a_1 (2 \cdot 5 - 1) = \frac{1}{2} a_1 (10 - 1) = \frac{9}{2} a_1 \]

2. **Understanding the Motion of Body B**:
- Body \( B \) starts from rest with an acceleration \( a_2 \) but starts 2 seconds after \( A \). Therefore, when \( A \) has been moving for 5 seconds, \( B \) has been moving for \( 5 - 2 = 3 \) seconds.
- The distance traveled by body \( B \) in the 3rd second can be calculated similarly:
\[ S_3 = u + \frac{1}{2} a (2n - 1) \]
- For body \( B \), since it starts from rest, \( u = 0 \) and for \( n = 3 \):
\[ S_3 = 0 + \frac{1}{2} a_2 (2 \cdot 3 - 1) = \frac{1}{2} a_2 (6 - 1) = \frac{5}{2} a_2 \]

3. **Setting the Distances Equal**:
- According to the problem, the distances traveled by both bodies in their respective times are equal:
\[ S_5 = S_3 \]
- Substituting the expressions we derived:
\[ \frac{9}{2} a_1 = \frac{5}{2} a_2 \]

4. **Solving for the Ratio**:
- To find the ratio \( \frac{a_1}{a_2} \), we can rearrange the equation:
\[ 9 a_1 = 5 a_2 \]
- Dividing both sides by \( 5 a_2 \):
\[ \frac{a_1}{a_2} = \frac{5}{9} \]

### Final Answer:
The ratio \( a_1 : a_2 \) is \( 5 : 9 \).