Position of a particle at any instant is given by `x = 3t^(2)+1` , where x is in m and t in sec. Its average velocity in the time interval `t = 2` sec to `t = 3` sec will be :

Explanation

To find the average velocity of the particle in the time interval from \( t = 2 \) seconds to \( t = 3 \) seconds, we can follow these steps:

### Step 1: Determine the position at \( t = 2 \) seconds
The position of the particle is given by the equation:
\[ x = 3t^2 + 1 \]
Substituting \( t = 2 \):
\[ x(2) = 3(2^2) + 1 = 3(4) + 1 = 12 + 1 = 13 \text{ m} \]

### Step 2: Determine the position at \( t = 3 \) seconds
Now, substituting \( t = 3 \):
\[ x(3) = 3(3^2) + 1 = 3(9) + 1 = 27 + 1 = 28 \text{ m} \]

### Step 3: Calculate the displacement
The displacement \( \Delta x \) over the interval from \( t = 2 \) seconds to \( t = 3 \) seconds is:
\[ \Delta x = x(3) - x(2) = 28 \text{ m} - 13 \text{ m} = 15 \text{ m} \]

### Step 4: Calculate the time interval
The time interval \( \Delta t \) is:
\[ \Delta t = t_2 - t_1 = 3 \text{ s} - 2 \text{ s} = 1 \text{ s} \]

### Step 5: Calculate the average velocity
The average velocity \( v_{avg} \) is given by the formula:
\[ v_{avg} = \frac{\Delta x}{\Delta t} \]
Substituting the values we found:
\[ v_{avg} = \frac{15 \text{ m}}{1 \text{ s}} = 15 \text{ m/s} \]

Thus, the average velocity of the particle in the time interval from \( t = 2 \) seconds to \( t = 3 \) seconds is:
\[ \boxed{15 \text{ m/s}} \]