Average velocity of a particle moving in a straight line, with constant acceleration a and initial velocity `u` in first `t` seconds is.

Options

A. ` u+ (1)/(2) at `
B. ` u+ at`
C. ` (u+ at)/(2) `
D. ` (u)/(2) `

Explanation

To find the average velocity of a particle moving in a straight line with constant acceleration, we can follow these steps:

### Step 1: Understand the parameters
We have:
- Initial velocity (u)
- Constant acceleration (a)
- Time duration (t)

### Step 2: Use the formula for average velocity
The average velocity (V_avg) of an object moving with constant acceleration can be calculated using the formula:
\[ V_{avg} = \frac{V_f + V_i}{2} \]
where \(V_f\) is the final velocity and \(V_i\) is the initial velocity.

### Step 3: Calculate the final velocity
The final velocity (V_f) after time t can be calculated using the equation of motion:
\[ V_f = u + at \]

### Step 4: Substitute the values into the average velocity formula
Now, substitute \(V_f\) and \(V_i\) into the average velocity formula:
\[ V_{avg} = \frac{(u + at) + u}{2} \]

### Step 5: Simplify the expression
Combine the terms in the numerator:
\[ V_{avg} = \frac{2u + at}{2} \]

### Step 6: Final expression for average velocity
Thus, the average velocity of the particle in the first t seconds is:
\[ V_{avg} = u + \frac{at}{2} \]

### Conclusion
The average velocity of the particle moving in a straight line with constant acceleration a and initial velocity u in the first t seconds is:
\[ V_{avg} = u + \frac{at}{2} \]

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