Following are four different relations about displacement, velocity and acceleration for the motion of a particle in general. Choose the incorrect one (s)

Options

A. `v_("av")=(1)/(2)[v(t_(1))+v(t_(2))]`
B. `v_("av")=(r(t_(2))-r(t_(1)))/(t_(2)-t_(1))`
C. `r=(1)/(2)(v(t_(2))-v(t_(1))) (t_(2)-t_(1))`
D. `a_("av")=(v(t_(2))-v(t_(1)))/(t_(2)-t_(1))`

Explanation

To solve the question of identifying the incorrect relation among the four given options regarding displacement, velocity, and acceleration, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Average Velocity**:
- The average velocity (\( V_{avg} \)) is defined as the total displacement divided by the total time taken.
- Mathematically, it can be expressed as:
\[ V_{avg} = \frac{\Delta r}{\Delta t} = \frac{r_{t2} - r_{t1}}{t2 - t1} \]
- This relation is correct and is typically found in the options.

2. **Understanding Average Acceleration**:
- The average acceleration (\( A_{avg} \)) is defined as the change in velocity divided by the change in time.
- Mathematically, it can be expressed as:
\[ A_{avg} = \frac{\Delta v}{\Delta t} = \frac{v_{t2} - v_{t1}}{t2 - t1} \]
- This relation is also correct and is typically found in the options.

3. **Analyzing the Relation for Displacement**:
- One of the options may state a relation like:
\[ r = \frac{1}{2} (v_{t2} - v_{t1})(t2 - t1) \]
- This relation is derived from the kinematic equations under uniform acceleration. It states that the displacement is equal to the average of the initial and final velocities multiplied by the time interval. However, this is only valid for uniform acceleration.

4. **Considering Non-Uniform Acceleration**:
- In the case of non-uniform acceleration, the average velocity is not simply the average of the initial and final velocities. Therefore, the relation mentioned in the previous step does not hold true for non-uniform acceleration.
- Hence, if the question does not specify that the acceleration is uniform, this relation can be considered incorrect.

5. **Identifying the Incorrect Option**:
- Based on the analysis, if one of the options states that:
\[ r = \frac{1}{2} (v_{t2} + v_{t1})(t2 - t1) \]
- This would be the incorrect option unless specified that the motion is under uniform acceleration.

### Conclusion:
- The incorrect relation among the options is the one that assumes the average velocity is the average of the initial and final velocities without specifying uniform acceleration.