Following are four different relation about displacement,velocity and acceleration for the motion of a particle in general.Choose the incorrect one (S).
a) `V_(av)=(1)/(2)[v(t_(1))+v(t_(2))]`
b)`V_(av)=(r(t_(2))-r(t_(2)))/(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))`

Options

A. a and b
B. a and d
C. b and c
D. a and c

Answer

To determine the incorrect relation among the four given options regarding displacement, velocity, and acceleration, let's analyze each option step by step.

### Step 1: Analyze Option (a)
**Relation:** \( V_{av} = \frac{1}{2} [v(t_1) + v(t_2)] \)

**Explanation:** This equation suggests that the average velocity is the arithmetic mean of the velocities at two different times \( t_1 \) and \( t_2 \). However, average velocity is defined as the total displacement divided by the total time taken. This relation is only valid if the motion is uniform (constant velocity). Therefore, this relation is **incorrect** in general.

### Step 2: Analyze Option (b)
**Relation:** \( V_{av} = \frac{r(t_2) - r(t_1)}{t_2 - t_1} \)

**Explanation:** This equation correctly defines the average velocity as the change in displacement divided by the change in time. This is a standard definition in physics and is **correct**.

### Step 3: Analyze Option (c)
**Relation:** \( r = \frac{1}{2} (v(t_2) - v(t_1))(t_2 - t_1) \)

**Explanation:** This equation suggests that displacement \( r \) is equal to half the product of the change in velocity and the time interval. However, displacement is actually calculated using average velocity, not instantaneous velocities. The correct relation should involve average velocity, not just the difference in velocities. Thus, this relation is also **incorrect**.

### Step 4: Analyze Option (d)
**Relation:** \( a_{av} = \frac{v(t_2) - v(t_1)}{t_2 - t_1} \)

**Explanation:** This equation correctly defines the average acceleration as the change in velocity divided by the change in time. This is a standard definition in physics and is **correct**.

### Conclusion
The incorrect relations are:
- **(a)** \( V_{av} = \frac{1}{2} [v(t_1) + v(t_2)] \)
- **(c)** \( r = \frac{1}{2} (v(t_2) - v(t_1))(t_2 - t_1) \)

### Summary of Incorrect Options
- **Incorrect Options:** (a) and (c)