For body moving with uniform acceleration `a`, initial and final velocities in a time interval `t` are `u` and `v` respectively. Then its average velocity in the time interval `t` is

Options

A. `(v+at)`
B. `(v-(at)/(2))`
C. `(v-at)`
D. `(u-(at)/(2))`

Explanation

To find the average velocity of a body moving with uniform acceleration, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Given Information**:
- The body has an initial velocity \( u \) and a final velocity \( v \).
- The acceleration is uniform, denoted as \( a \).
- The time interval during which this motion occurs is \( t \).

2. **Formula for Average Velocity**:
- The average velocity \( v_{\text{avg}} \) can be defined as the total distance traveled divided by the total time taken:
\[ v_{\text{avg}} = \frac{s}{t} \]
where \( s \) is the distance traveled during the time interval \( t \).

3. **Distance Traveled with Uniform Acceleration**:
- The formula for distance \( s \) when an object is moving with uniform acceleration is given by:
\[ s = ut + \frac{1}{2} a t^2 \]

4. **Substituting for Average Velocity**:
- Substitute the expression for \( s \) into the average velocity formula:
\[ v_{\text{avg}} = \frac{ut + \frac{1}{2} a t^2}{t} \]
- Simplifying this gives:
\[ v_{\text{avg}} = u + \frac{1}{2} a t \]

5. **Relate Final Velocity to Initial Velocity**:
- From the equation of motion, we know:
\[ v = u + at \]
- Rearranging this gives:
\[ u = v - at \]

6. **Substituting \( u \) in Average Velocity Formula**:
- Now substitute \( u \) in the average velocity formula:
\[ v_{\text{avg}} = (v - at) + \frac{1}{2} a t \]
- Simplifying this gives:
\[ v_{\text{avg}} = v - at + \frac{1}{2} a t = v - \frac{1}{2} a t \]

7. **Final Expression for Average Velocity**:
- Thus, the average velocity can be expressed as:
\[ v_{\text{avg}} = v - \frac{1}{2} a t \]

### Conclusion:
The average velocity \( v_{\text{avg}} \) of the body moving with uniform acceleration over the time interval \( t \) is given by:
\[ v_{\text{avg}} = u + \frac{1}{2} a t \]
or equivalently,
\[ v_{\text{avg}} = v - \frac{1}{2} a t \]