A car starts moving on a straight road. It makes the entire journey in three parts. The total time of journey is 25 s. For first part it accelerates at a constant rate `5m//s^(2)`. For second part it moves with a constant speed. For the third part it retards at constant rate `5m//s^(2)` to come to rest.
Let `t_(1)` be the time for first part, `t_(2)` be the time for 2nd part then correct relation is

Options

A. `t_(2)=25-t_(1)`
B. `t_(2)=25-2t_(1)`
C. `2t_(1)=25-t_(2)`
D. None of these

Answer

To solve the problem, we will analyze the motion of the car in three parts: acceleration, constant speed, and deceleration. We will use the equations of motion to derive the relationships between the times taken in each part.

### Step-by-Step Solution:

1. **Understanding the Motion**:
- The car accelerates for time \( t_1 \) at a rate of \( 5 \, \text{m/s}^2 \).
- It then moves with a constant speed for time \( t_2 \).
- Finally, it decelerates at a rate of \( 5 \, \text{m/s}^2 \) for time \( t_3 \) until it comes to rest.

2. **Using the First Equation of Motion**:
- For the first part (acceleration):
\[ v_0 = u + at \]
Here, \( u = 0 \) (initial velocity), \( a = 5 \, \text{m/s}^2 \), and \( t = t_1 \).
\[ v_0 = 0 + 5t_1 \implies v_0 = 5t_1 \quad \text{(Equation 1)} \]

3. **Using the First Equation of Motion for Deceleration**:
- For the third part (deceleration):
\[ v = u + at \]
Here, \( v = 0 \) (final velocity), \( u = v_0 \), \( a = -5 \, \text{m/s}^2 \), and \( t = t_3 \).
\[ 0 = v_0 - 5t_3 \implies v_0 = 5t_3 \quad \text{(Equation 2)} \]

4. **Equating the Two Expressions for \( v_0 \)**:
- From Equations 1 and 2:
\[ 5t_1 = 5t_3 \implies t_1 = t_3 \quad \text{(Equation 3)} \]

5. **Total Time of Journey**:
- The total time for the journey is given as:
\[ t_1 + t_2 + t_3 = 25 \, \text{s} \]
- Substituting \( t_3 \) with \( t_1 \) from Equation 3:
\[ t_1 + t_2 + t_1 = 25 \implies 2t_1 + t_2 = 25 \]

6. **Final Relation**:
- Rearranging gives:
\[ t_2 = 25 - 2t_1 \]

### Conclusion:
The correct relation between the times is:
\[ t_2 = 25 - 2t_1 \]