Graphs are very useful to represent a physical situation. Various quantities can be easily represented on graphs and other quantites can be determined from the graph. For example the slope of velocity-time graph represents instantaneous acceleration. For a motion with constant acceleration slope of velocity-time graph is constant. If acceleration is changing with time, slope will change and thus velocity-time graph will be a non-linear curve. Further the area of velocity-time graph gives displacement.
The correct relation between `v_(0),a_(1),a_(2) and t_(0)` is

Options

A. `v_(0)=((a_(1)a_(2))/(a_(1)+a_(2)))t_(0)^(2)`
B. `v_(0)=((a_(1)a_(2))/(a_(1)+a_(2)))t_(0)`
C. `v_(0)=((a_(1)+a_(2))/(2))t_(0)`
D. `v_(0)=sqrt(a_(1)a_(2)t_(0))`

Answer

To find the correct relation between \( V_0 \), \( A_1 \), \( A_2 \), and \( T_0 \), we can analyze the velocity-time graph and the corresponding accelerations.

### Step-by-Step Solution:

1. **Understanding the Graphs**:
- We have a velocity-time graph where the velocity increases initially and then decreases.
- The initial velocity at time \( t = 0 \) is \( V_0 \).
- The slope of the velocity-time graph gives us the acceleration.

2. **Identifying Accelerations**:
- When the velocity is increasing, the acceleration is \( A_1 \).
- When the velocity is decreasing, the acceleration is \( A_2 \).
- The relation for acceleration can be expressed as:
\[ A_1 = \frac{V_0}{T_0} \quad \text{(1)} \]
\[ A_2 = \frac{V_0}{T_1} \quad \text{(2)} \]
where \( T_0 \) is the time for which the velocity is increasing, and \( T_1 \) is the time for which the velocity is decreasing until it becomes zero.

3. **Total Time**:
- The total time \( T \) for the motion is given by:
\[ T = T_0 + T_1 \quad \text{(3)} \]

4. **Expressing \( T_1 \)**:
- From equation (2), we can express \( T_1 \) as:
\[ T_1 = \frac{V_0}{A_2} \quad \text{(4)} \]

5. **Substituting \( T_1 \) in Total Time**:
- Substitute equation (4) into equation (3):
\[ T = T_0 + \frac{V_0}{A_2} \]

6. **Substituting \( T_0 \)**:
- From equation (1), we can express \( T_0 \) as:
\[ T_0 = \frac{V_0}{A_1} \quad \text{(5)} \]
- Substitute equation (5) into the total time equation:
\[ T = \frac{V_0}{A_1} + \frac{V_0}{A_2} \]

7. **Finding a Common Denominator**:
- Combine the fractions:
\[ T = V_0 \left( \frac{1}{A_1} + \frac{1}{A_2} \right) = V_0 \cdot \frac{A_1 + A_2}{A_1 A_2} \]

8. **Rearranging for \( V_0 \)**:
- Rearranging gives us:
\[ V_0 = T \cdot \frac{A_1 A_2}{A_1 + A_2} \]

### Final Relation:
Thus, the correct relation between \( V_0 \), \( A_1 \), \( A_2 \), and \( T \) is:
\[ V_0 = T \cdot \frac{A_1 A_2}{A_1 + A_2} \]