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 `5th` 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 accelerations \( a_1 \) and \( a_2 \) for two bodies \( A \) and \( B \) that travel equal distances during specific time intervals. Here’s a step-by-step solution:

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

### Step 2: Understand the Motion of Body B
Body \( B \) starts from rest with an acceleration \( a_2 \) but starts 2 seconds after body \( A \). Therefore, when \( A \) has traveled for 5 seconds, \( B \) has only traveled for 3 seconds. The distance traveled by body \( B \) in the \( 3^{rd} \) second can be calculated using the same formula:
\[ S_n = u + \frac{1}{2} a t^2 \]
For body \( B \):
\[ S_3 = 0 + \frac{1}{2} a_2 (2 \cdot 3 - 1) = \frac{1}{2} a_2 (6 - 1) = \frac{5}{2} a_2 \]

### Step 3: Set the Distances Equal
According to the problem, the distances traveled by both bodies in their respective time intervals are equal:
\[ S_5 \text{ (for A)} = S_3 \text{ (for B)} \]
Thus, we have:
\[ \frac{9}{2} a_1 = \frac{5}{2} a_2 \]

### Step 4: Solve for the Ratio of Accelerations
To find the ratio \( \frac{a_1}{a_2} \), we can rearrange the equation:
\[ 9 a_1 = 5 a_2 \]
Dividing both sides by \( 5 a_2 \):
\[ \frac{a_1}{a_2} = \frac{5}{9} \]

### Conclusion
The ratio of the accelerations \( a_1 : a_2 \) is:
\[ \boxed{\frac{5}{9}} \]