A particle moves in a straight line, its position (in m) as function of time is given by `x = (at^2 + b)`.
What is average velocity in time interval `t = 3 sec` to `t = 5 sec` ? (Where `a` and `b` are constant and `a = 1 m//s^2, b = 1 m`.)

Options

A. `8 m//s`
B. `5 m//s`
C. `10 m//s`
D. `12 m//s`

Explanation

To solve the problem of finding the average velocity of a particle moving in a straight line, we will follow these steps:

### Step 1: Understand the position function
The position of the particle as a function of time is given by:
\[ x(t) = at^2 + b \]
where \( a = 1 \, \text{m/s}^2 \) and \( b = 1 \, \text{m} \).

### Step 2: Calculate the position at \( t = 3 \) seconds
Substituting \( t = 3 \) seconds into the position function:
\[ x(3) = a(3^2) + b = 1(3^2) + 1 = 1(9) + 1 = 9 + 1 = 10 \, \text{m} \]

### Step 3: Calculate the position at \( t = 5 \) seconds
Now, substituting \( t = 5 \) seconds into the position function:
\[ x(5) = a(5^2) + b = 1(5^2) + 1 = 1(25) + 1 = 25 + 1 = 26 \, \text{m} \]

### Step 4: Calculate the displacement
The displacement (\( \Delta x \)) during the time interval from \( t = 3 \) seconds to \( t = 5 \) seconds is given by:
\[ \Delta x = x(5) - x(3) = 26 \, \text{m} - 10 \, \text{m} = 16 \, \text{m} \]

### Step 5: Calculate the time interval
The time interval (\( \Delta t \)) is:
\[ \Delta t = 5 \, \text{s} - 3 \, \text{s} = 2 \, \text{s} \]

### Step 6: Calculate the average velocity
The average velocity (\( v_{avg} \)) is given by the formula:
\[ v_{avg} = \frac{\Delta x}{\Delta t} = \frac{16 \, \text{m}}{2 \, \text{s}} = 8 \, \text{m/s} \]

### Final Answer
The average velocity of the particle in the time interval from \( t = 3 \) seconds to \( t = 5 \) seconds is:
\[ \boxed{8 \, \text{m/s}} \]

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