A car starts from rest moves with uniform acceleration `a_(1)` for `t_(1)` second and then retards uniformly at a rate `a_(2)` for `t_(2)` second. Then `t_(1)//t_(2)` is equal to

Options

A. `a_(1)//a_(2)`
B. `a_(2)//a_(1)`
C. `a_(2)^(2)//a_(1)`
D. `2a_(1)//a_(2)`

Explanation

To solve the problem step by step, we need to analyze the motion of the car during its acceleration and deceleration phases.

### Step 1: Understand the motion phases
The car starts from rest, accelerates uniformly for time \( t_1 \) with acceleration \( a_1 \), then decelerates uniformly for time \( t_2 \) with deceleration \( a_2 \).

### Step 2: Apply the equations of motion
1. **During acceleration (from rest)**:
- Initial velocity \( u = 0 \)
- Final velocity \( v \) after time \( t_1 \):
\[ v = u + a_1 t_1 = 0 + a_1 t_1 = a_1 t_1 \]
- Distance covered during acceleration \( s_1 \):
\[ s_1 = ut + \frac{1}{2} a_1 t_1^2 = 0 + \frac{1}{2} a_1 t_1^2 = \frac{1}{2} a_1 t_1^2 \]

2. **During deceleration**:
- Initial velocity \( u' = v = a_1 t_1 \)
- Final velocity \( v' = 0 \) (the car comes to rest)
- Using the equation \( v' = u' - a_2 t_2 \):
\[ 0 = a_1 t_1 - a_2 t_2 \]
- Rearranging gives:
\[ a_1 t_1 = a_2 t_2 \]

### Step 3: Find the ratio \( \frac{t_1}{t_2} \)
From the equation \( a_1 t_1 = a_2 t_2 \), we can express the ratio \( \frac{t_1}{t_2} \):
\[ \frac{t_1}{t_2} = \frac{a_2}{a_1} \]

### Final Answer
Thus, the ratio \( \frac{t_1}{t_2} \) is equal to:
\[ \frac{t_1}{t_2} = \frac{a_2}{a_1} \]

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