A car, starting from rest, has a constant acceleration for a time interval ` t_1` during which it covers a distance In the next time interval the car has a constant retardation and comes to rest after covering a distance in time Which of the following relations is correct?

Options

A. ` (a_1) /(a_2) =(s_1)/( S_2) =(t_1)/(t_2) `
B. ` (a_1)/(a_2) =(s_2)/(s_1) =(t_1)/(t_2)`
C. ` (a_1)/(a_2) =(s_1)/(s_2) =(t_2)/(t_1)`
D. ` (a_1)/(a_2)=(s_2)/(s_1) =(t_2)/(t_1)`

Explanation

To solve the problem, we need to analyze the motion of the car in two phases: the first phase with constant acceleration and the second phase with constant retardation.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The car starts from rest, meaning its initial velocity \( u = 0 \).
- During the first time interval \( t_1 \), the car accelerates uniformly and covers a distance \( s_1 \).
- During the second time interval \( t_2 \), the car decelerates uniformly and comes to rest, covering a distance \( s_2 \).

2. **Using Kinematic Equations**:
- For the first phase (acceleration):
- The final velocity \( v \) after time \( t_1 \) can be expressed as:
\[ v = u + a_1 t_1 = 0 + a_1 t_1 = a_1 t_1 \]
- The distance covered \( s_1 \) can be expressed as:
\[ s_1 = ut + \frac{1}{2} a_1 t_1^2 = 0 + \frac{1}{2} a_1 t_1^2 = \frac{1}{2} a_1 t_1^2 \]

- For the second phase (retardation):
- The car comes to rest, so the final velocity is \( 0 \) and the initial velocity is \( v \):
\[ 0 = v - a_2 t_2 \implies a_2 = \frac{v}{t_2} = \frac{a_1 t_1}{t_2} \]
- The distance covered \( s_2 \) during this phase can be expressed as:
\[ s_2 = vt_2 - \frac{1}{2} a_2 t_2^2 = a_1 t_1 t_2 - \frac{1}{2} \left(\frac{a_1 t_1}{t_2}\right) t_2^2 = a_1 t_1 t_2 - \frac{1}{2} a_1 t_1 t_2 = \frac{1}{2} a_1 t_1 t_2 \]

3. **Finding Relationships**:
- From the expressions for \( s_1 \) and \( s_2 \):
\[ s_1 = \frac{1}{2} a_1 t_1^2 \]
\[ s_2 = \frac{1}{2} a_1 t_1 t_2 \]
- Dividing \( s_2 \) by \( s_1 \):
\[ \frac{s_2}{s_1} = \frac{\frac{1}{2} a_1 t_1 t_2}{\frac{1}{2} a_1 t_1^2} = \frac{t_2}{t_1} \]
- Therefore, we can conclude:
\[ \frac{a_1}{a_2} = \frac{t_2}{t_1} \quad \text{and} \quad \frac{s_1}{s_2} = \frac{t_2}{t_1} \]

### Conclusion:
The correct relation is:
\[ \frac{a_1}{a_2} = \frac{s_2}{s_1} = \frac{t_2}{t_1} \]