A particle moves along x-axis as `x=4(t-2)+a(t-2)^2`
Which of the following is true?

Options

A. The initial velocity of particle is 4
B. the acceleration of particle is 2a
C. The particle is at origin at `t=0`
D. none of these

Explanation

To solve the problem, we need to analyze the motion of the particle described by the equation:

\[ x = 4(t - 2) + a(t - 2)^2 \]

### Step 1: Find the initial velocity of the particle
To find the initial velocity, we need to differentiate the position function \( x \) with respect to time \( t \):

\[ v = \frac{dx}{dt} = \frac{d}{dt}[4(t - 2) + a(t - 2)^2] \]

Using the chain rule, we differentiate:

1. The derivative of \( 4(t - 2) \) is \( 4 \).
2. The derivative of \( a(t - 2)^2 \) is \( 2a(t - 2) \).

Thus, we have:

\[ v = 4 + 2a(t - 2) \]

### Step 2: Evaluate the initial velocity at \( t = 0 \)
Now, we substitute \( t = 0 \) into the velocity equation:

\[ v(0) = 4 + 2a(0 - 2) = 4 - 4a \]

### Step 3: Check if the initial velocity is equal to 4
For the initial velocity to be 4, we set the equation:

\[ 4 - 4a = 4 \]

Solving for \( a \):

\[ -4a = 0 \implies a = 0 \]

Thus, the initial velocity is not necessarily 4 unless \( a = 0 \).

### Step 4: Find the acceleration of the particle
To find the acceleration, we differentiate the velocity function:

\[ a = \frac{d^2x}{dt^2} = \frac{d}{dt}[4 + 2a(t - 2)] \]

The derivative of \( 4 \) is \( 0 \), and the derivative of \( 2a(t - 2) \) is \( 2a \). Therefore:

\[ a = 2a \]

### Step 5: Conclusion about the acceleration
The acceleration of the particle is constant and equal to \( 2a \).

### Step 6: Check if the particle is at the origin at \( t = 0 \)
To check if the particle is at the origin, we evaluate \( x \) at \( t = 0 \):

\[ x(0) = 4(0 - 2) + a(0 - 2)^2 = 4(-2) + a(4) = -8 + 4a \]

For the particle to be at the origin, we set:

\[ -8 + 4a = 0 \implies 4a = 8 \implies a = 2 \]

### Final Summary
- The initial velocity is \( 4 - 4a \).
- The acceleration is \( 2a \).
- The particle is at the origin at \( t = 0 \) if \( a = 2 \).

### Conclusion
The correct statement from the options provided is that the acceleration of the particle is \( 2a \).