A point moves with uniform acceleration and `v_(1), v_(2)`, and `v_(3)` denote the average velocities in the three successive intervals of time `t_(1).t_(2)`, and `t_(3)` Which of the following Relations is correct?.

Options

A. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(1)-t_(2):(t_(2)+t_(3)`.
B. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(2)-t_(2):(t_(2)+t_(3)`
C. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(1)-t_(2):(t_(2)+t_(3)`
D. `(v_(1)-v_(2)):(v_(2)-v_(3)=(t_(1)-t_(2):(t_(2)+t_(3)`

Explanation

To solve the problem, we need to analyze the average velocities during the three successive time intervals of motion with uniform acceleration. Let's denote the average velocities in the intervals \( t_1, t_2, \) and \( t_3 \) as \( v_1, v_2, \) and \( v_3 \) respectively.

### Step-by-Step Solution:

1. **Understanding Average Velocity**:
The average velocity over a time interval is given by the formula:
\[ v_{\text{avg}} = \frac{u + v}{2} \]
where \( u \) is the initial velocity and \( v \) is the final velocity.

2. **Calculate Final Velocities**:
Using the first equation of motion:
- After time \( t_1 \):
\[ v_{11} = u + at_1 \]
- After time \( t_1 + t_2 \):
\[ v_{22} = u + a(t_1 + t_2) = u + at_1 + at_2 \]
- After time \( t_1 + t_2 + t_3 \):
\[ v_{33} = u + a(t_1 + t_2 + t_3) = u + at_1 + at_2 + at_3 \]

3. **Calculate Average Velocities**:
Now we can calculate the average velocities:
- For the first interval \( t_1 \):
\[ v_1 = \frac{u + v_{11}}{2} = \frac{u + (u + at_1)}{2} = \frac{2u + at_1}{2} = u + \frac{at_1}{2} \]
- For the second interval \( t_2 \):
\[ v_2 = \frac{v_{11} + v_{22}}{2} = \frac{(u + at_1) + (u + at_1 + at_2)}{2} = \frac{2u + 2at_1 + at_2}{2} = u + at_1 + \frac{at_2}{2} \]
- For the third interval \( t_3 \):
\[ v_3 = \frac{v_{22} + v_{33}}{2} = \frac{(u + at_1 + at_2) + (u + at_1 + at_2 + at_3)}{2} = \frac{2u + 2at_1 + 2at_2 + at_3}{2} = u + at_1 + at_2 + \frac{at_3}{2} \]

4. **Finding the Relations**:
Now we need to find the relations between \( v_1, v_2, \) and \( v_3 \):
- Calculate \( v_1 - v_2 \):
\[ v_1 - v_2 = \left(u + \frac{at_1}{2}\right) - \left(u + at_1 + \frac{at_2}{2}\right) = \frac{at_1}{2} - at_1 - \frac{at_2}{2} = -\frac{at_1}{2} - \frac{at_2}{2} = -\frac{a(t_1 + t_2)}{2} \]
- Calculate \( v_2 - v_3 \):
\[ v_2 - v_3 = \left(u + at_1 + \frac{at_2}{2}\right) - \left(u + at_1 + at_2 + \frac{at_3}{2}\right) = \frac{at_2}{2} - at_2 - \frac{at_3}{2} = -\frac{at_2}{2} - \frac{at_3}{2} = -\frac{a(t_2 + t_3)}{2} \]

5. **Forming the Final Relation**:
Now we can form the ratio:
\[ \frac{v_1 - v_2}{v_2 - v_3} = \frac{-\frac{a(t_1 + t_2)}{2}}{-\frac{a(t_2 + t_3)}{2}} = \frac{t_1 + t_2}{t_2 + t_3} \]

### Conclusion:
The correct relation is:
\[ \frac{v_1 - v_2}{v_2 - v_3} = \frac{t_1 + t_2}{t_2 + t_3} \]