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. 0.21458333333333
B. 0.21319444444444
C. 0.37847222222222
D. 0.37986111111111

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 start from rest and travel equal distances in the 5th second after the start of \( A \).

### Step-by-Step Solution:

1. **Understanding the Problem**:
- Body \( A \) starts from rest with acceleration \( a_1 \) at \( t = 0 \).
- Body \( B \) starts from rest with acceleration \( a_2 \) at \( t = 2 \) seconds.
- We need to find the ratio \( \frac{a_1}{a_2} \) given that both bodies travel equal distances in the 5th second after the start of \( A \).

2. **Distance Covered in the nth Second**:
- The formula for the distance covered by an object starting from rest in the nth second is given by:
\[ s_n = \frac{1}{2} a (2n - 1) \]
- Here, \( s_n \) is the distance covered in the nth second, and \( a \) is the acceleration.

3. **Distance Covered by Body A in the 5th Second**:
- For body \( A \) (where \( n = 5 \)):
\[ s_A = \frac{1}{2} a_1 (2 \times 5 - 1) = \frac{1}{2} a_1 (10 - 1) = \frac{9}{2} a_1 \]

4. **Distance Covered by Body B in the 3rd Second**:
- Body \( B \) starts at \( t = 2 \) seconds, so by the time it reaches the 5th second, it has been moving for \( 5 - 2 = 3 \) seconds.
- For body \( B \) (where \( n = 3 \)):
\[ s_B = \frac{1}{2} a_2 (2 \times 3 - 1) = \frac{1}{2} a_2 (6 - 1) = \frac{5}{2} a_2 \]

5. **Setting Distances Equal**:
- Since both bodies travel equal distances in the 5th second:
\[ s_A = s_B \]
\[ \frac{9}{2} a_1 = \frac{5}{2} a_2 \]

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

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