A particle is moving along X-axis according to the following equation: `x=u(t-4)+a(t-3)^(2)`
All terms in above equation are measured in MKS system

Options

A. Acceleration of the particle is a.
B. Acceleration of the particle is 2a.
C. Velocity of particle at t=3 s is u.
D. At t=0 particle is at the origin.

Explanation

To solve the problem step by step, we will analyze the given equation of motion and derive the required quantities: acceleration and velocity.

### Given Equation:
The equation of motion for the particle is given as:
\[ x = u(t - 4) + a(t - 3)^2 \]

### Step 1: Find the Velocity
Velocity \( v \) is defined as the rate of change of displacement with respect to time. We can find the velocity by differentiating the displacement equation with respect to time \( t \).

#### Differentiate the equation:
\[ v = \frac{dx}{dt} \]
Using the product rule and chain rule:
1. Differentiate \( u(t - 4) \):
\[ \frac{d}{dt}[u(t - 4)] = u \]
2. Differentiate \( a(t - 3)^2 \):
\[ \frac{d}{dt}[a(t - 3)^2] = a \cdot 2(t - 3) = 2a(t - 3) \]

Combining these results, we have:
\[ v = u + 2a(t - 3) \]

### Step 2: Find the Velocity at \( t = 3 \) seconds
Now, we need to find the velocity at \( t = 3 \) seconds:
\[ v(t = 3) = u + 2a(3 - 3) \]
\[ v(t = 3) = u + 2a(0) \]
\[ v(t = 3) = u \]

### Step 3: Find the Acceleration
Acceleration \( a \) is defined as the rate of change of velocity with respect to time. We can find acceleration by differentiating the velocity equation with respect to time \( t \).

#### Differentiate the velocity equation:
\[ a = \frac{dv}{dt} \]
From our previous velocity equation:
\[ v = u + 2a(t - 3) \]
Differentiating:
1. The term \( u \) is constant, so its derivative is \( 0 \).
2. Differentiate \( 2a(t - 3) \):
\[ \frac{d}{dt}[2a(t - 3)] = 2a \]

Thus, we have:
\[ a = 2a \]

### Conclusion
- The velocity of the particle at \( t = 3 \) seconds is \( u \).
- The acceleration of the particle is \( 2a \).

### Final Answers:
- **Velocity at \( t = 3 \) seconds:** \( u \)
- **Acceleration:** \( 2a \)