The position of an object moving along x-axis is given by x`=a+bt^(2)` where a =8.5 m, b=2.5 `ms^(-2)` and t is measured in seconds.Then which of the following is true ?

Options

A. Velocity at t=2 sec is zero
B. Average velocity between t=2,t=4 sec is 15 m/s
C. Velocity at t=4 sec is 10 m/s
D. All the above are true

Answer

To solve the problem, we need to analyze the position function given and derive the necessary quantities such as velocity and average velocity.

### Given:
- Position function: \( x = a + bt^2 \)
- Constants: \( a = 8.5 \, \text{m} \), \( b = 2.5 \, \text{ms}^{-2} \)

### Step 1: Finding the Velocity Function
The velocity \( v \) is the derivative of the position \( x \) with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(a + bt^2) = 0 + 2bt = 2bt \]
Substituting the value of \( b \):
\[ v = 2 \times 2.5 \times t = 5t \, \text{m/s} \]

### Step 2: Calculate Velocity at \( t = 2 \) seconds
Now, we will find the velocity at \( t = 2 \) seconds:
\[ v(2) = 5 \times 2 = 10 \, \text{m/s} \]
This shows that the first option stating that the velocity at \( t = 2 \) seconds is 0 is incorrect.

### Step 3: Finding the Average Velocity between \( t = 2 \) seconds and \( t = 4 \) seconds
To find the average velocity, we first need to calculate the positions at \( t = 2 \) and \( t = 4 \) seconds.

#### Position at \( t = 2 \) seconds:
\[ x(2) = a + b(2^2) = 8.5 + 2.5 \times 4 = 8.5 + 10 = 18.5 \, \text{m} \]

#### Position at \( t = 4 \) seconds:
\[ x(4) = a + b(4^2) = 8.5 + 2.5 \times 16 = 8.5 + 40 = 48.5 \, \text{m} \]

#### Average Velocity:
The average velocity \( v_{avg} \) between \( t = 2 \) and \( t = 4 \) seconds is given by:
\[ v_{avg} = \frac{x(4) - x(2)}{4 - 2} = \frac{48.5 - 18.5}{2} = \frac{30}{2} = 15 \, \text{m/s} \]
This confirms that the second option stating the average velocity between \( t = 2 \) and \( t = 4 \) seconds is 15 m/s is correct.

### Step 4: Calculate Velocity at \( t = 4 \) seconds
Now we will find the velocity at \( t = 4 \) seconds:
\[ v(4) = 5 \times 4 = 20 \, \text{m/s} \]
This shows that the third option stating that the velocity at \( t = 4 \) seconds is 10 m/s is incorrect.

### Conclusion
- The first option is incorrect (velocity at \( t = 2 \) is not 0).
- The second option is correct (average velocity between \( t = 2 \) and \( t = 4 \) is 15 m/s).
- The third option is incorrect (velocity at \( t = 4 \) is not 10 m/s).
- The fourth option is incorrect (not all options are true).

Thus, the correct answer is that the second option is true.

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