A particle moves along the X-axis as `x=u(t-2 s)+a(t-2 s)^2`.

Options

A. the initial velocity of the particle is u
B. the acceleration of the particle is a
C. the acceleration of the particle is 2a
D. at t=2 s particle is at the origin.

Answer

To solve the problem step by step, we need to analyze the given position function of the particle and derive the necessary quantities: initial velocity, acceleration, and position at a specific time.

### Step 1: Understand the Position Function
The position of the particle is given by:
\[ x = u(t - 2 \, \text{s}) + a(t - 2 \, \text{s})^2 \]

### Step 2: Find the Velocity
To find the velocity, we differentiate the position function with respect to time \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}[u(t - 2) + a(t - 2)^2] \]

Using the product and chain rules:
1. The derivative of \( u(t - 2) \) is \( u \).
2. The derivative of \( a(t - 2)^2 \) is \( 2a(t - 2) \).

Thus, the velocity \( v \) becomes:
\[ v = u + 2a(t - 2) \]

### Step 3: Find the Acceleration
To find the acceleration, we differentiate the velocity with respect to time \( t \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}[u + 2a(t - 2)] \]

The derivative of \( u \) (a constant) is 0, and the derivative of \( 2a(t - 2) \) is \( 2a \). Therefore, the acceleration \( a \) is:
\[ a = 2a \]

### Step 4: Check the Position at \( t = 2 \, \text{s} \)
We need to find the position of the particle at \( t = 2 \, \text{s} \):
\[ x(2) = u(2 - 2) + a(2 - 2)^2 \]
This simplifies to:
\[ x(2) = u(0) + a(0)^2 = 0 + 0 = 0 \]

### Step 5: Determine Initial Velocity
To find the initial velocity at \( t = 0 \):
\[ v(0) = u + 2a(0 - 2) \]
This simplifies to:
\[ v(0) = u - 4a \]

### Summary of Results
1. The initial velocity is \( u - 4a \), which is not equal to \( u \).
2. The acceleration is \( 2a \).
3. At \( t = 2 \, \text{s} \), the particle is at the origin (position \( x = 0 \)).

### Conclusion
- The initial velocity of the particle is not \( u \) (it is \( u - 4a \)).
- The acceleration of the particle is \( 2a \).
- At \( t = 2 \, \text{s} \), the particle is at the origin.