An object starts from rest at the origin and moves along the `x` axis with a constant acceleration of `4m//s^(2)`. Its average velocity as it goes from `x=2` m to `x=8` m is :

Options

A. 2 m/s
B. 3 m/s
C. 5 m/s
D. 6 m/s

Answer

To find the average velocity of an object moving from \( x = 2 \, \text{m} \) to \( x = 8 \, \text{m} \) with a constant acceleration of \( 4 \, \text{m/s}^2 \), we can follow these steps:

### Step 1: Identify the initial conditions
The object starts from rest, so the initial velocity \( u = 0 \). The acceleration \( a = 4 \, \text{m/s}^2 \).

### Step 2: Use the equations of motion to find velocities at points B and C
- **Point B (at \( x = 2 \, \text{m} \))**:
We can use the third equation of motion:
\[ v^2 = u^2 + 2as \]
where \( s \) is the displacement from the origin to point B. Here, \( s = 2 \, \text{m} \):
\[ v_B^2 = 0^2 + 2 \times 4 \times 2 \]
\[ v_B^2 = 16 \implies v_B = \sqrt{16} = 4 \, \text{m/s} \]

- **Point C (at \( x = 8 \, \text{m} \))**:
Now, we find the velocity at point C using the same equation, where \( s = 8 \, \text{m} \):
\[ v_C^2 = 0^2 + 2 \times 4 \times 8 \]
\[ v_C^2 = 64 \implies v_C = \sqrt{64} = 8 \, \text{m/s} \]

### Step 3: Calculate the average velocity
The average velocity \( v_{\text{avg}} \) between two points can be calculated as:
\[ v_{\text{avg}} = \frac{v_B + v_C}{2} \]
Substituting the values we found:
\[ v_{\text{avg}} = \frac{4 + 8}{2} = \frac{12}{2} = 6 \, \text{m/s} \]

### Conclusion
The average velocity of the object as it moves from \( x = 2 \, \text{m} \) to \( x = 8 \, \text{m} \) is \( 6 \, \text{m/s} \).

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