A point moves with a uniform acceleration and ` v_1 ,v_2, v_3` denote the average velociies in the three succellive intervals of time `t _1` ,` t_2` and `t_3`. Find the ratio of ( `v_1` - `v_2`) and ( `v_2` - `v_3`).

Options

A. `t_1 - t_2/t_2 - t_3`
B. `t_1 - t_3/t_2 + t_3`
C. `t_1 + t_2/t_2 - t_3`
D. `t_1 + t_2/t_2 + t_3`

Explanation

To solve the problem, we need to find the ratio of \( (v_1 - v_2) \) and \( (v_2 - v_3) \) where \( v_1, v_2, v_3 \) are the average velocities during three successive time intervals \( t_1, t_2, t_3 \) under uniform acceleration.

### Step-by-Step Solution:

1. **Define Average Velocities**:
- The average velocity \( v_1 \) during the first interval \( t_1 \) is given by:
\[ v_1 = \frac{u + u_1}{2} \]
- The average velocity \( v_2 \) during the second interval \( t_2 \) is:
\[ v_2 = \frac{u_1 + u_2}{2} \]
- The average velocity \( v_3 \) during the third interval \( t_3 \) is:
\[ v_3 = \frac{u_2 + u_3}{2} \]

2. **Use Equations of Motion**:
- From the equations of motion, we can express \( u_1, u_2, \) and \( u_3 \):
- \( u_1 = u + a t_1 \)
- \( u_2 = u_1 + a t_2 = u + a t_1 + a t_2 = u + a(t_1 + t_2) \)
- \( u_3 = u_2 + a t_3 = u + a(t_1 + t_2) + a t_3 = u + a(t_1 + t_2 + t_3) \)

3. **Substitute into Average Velocities**:
- Substitute \( u_1, u_2, u_3 \) back into the expressions for \( v_1, v_2, v_3 \):
- \( v_1 = \frac{u + (u + a t_1)}{2} = \frac{2u + a t_1}{2} = u + \frac{a t_1}{2} \)
- \( v_2 = \frac{(u + a t_1) + (u + a(t_1 + t_2))}{2} = \frac{2u + 2a t_1 + a t_2}{2} = u + a t_1 + \frac{a t_2}{2} \)
- \( v_3 = \frac{(u + a(t_1 + t_2)) + (u + a(t_1 + t_2 + t_3))}{2} = \frac{2u + 2a(t_1 + t_2) + a t_3}{2} = u + a(t_1 + t_2) + \frac{a t_3}{2} \)

4. **Calculate \( v_1 - v_2 \) and \( v_2 - v_3 \)**:
- Calculate \( v_1 - v_2 \):
\[ v_1 - v_2 = \left(u + \frac{a t_1}{2}\right) - \left(u + a t_1 + \frac{a t_2}{2}\right) = \frac{a t_1}{2} - a t_1 - \frac{a t_2}{2} = -\frac{a t_1}{2} - \frac{a t_2}{2} = -\frac{a(t_1 + t_2)}{2} \]
- Calculate \( v_2 - v_3 \):
\[ v_2 - v_3 = \left(u + a t_1 + \frac{a t_2}{2}\right) - \left(u + a(t_1 + t_2) + \frac{a t_3}{2}\right) = a t_1 + \frac{a t_2}{2} - a(t_1 + t_2) - \frac{a t_3}{2} \]
Simplifying gives:
\[ = a t_1 + \frac{a t_2}{2} - a t_1 - a t_2 - \frac{a t_3}{2} = -\frac{a t_2}{2} - \frac{a t_3}{2} = -\frac{a(t_2 + t_3)}{2} \]

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

### Final Result:
The ratio of \( (v_1 - v_2) \) to \( (v_2 - v_3) \) is:
\[ \frac{v_1 - v_2}{v_2 - v_3} = \frac{t_1 + t_2}{t_2 + t_3} \]