The equation of one dimensional motion of the particle is described in column I. At `t=0`, particle is at origin and at rest. Match the column I with the statement in Column II.
`{:(,"Column I",,,"Column II"),((A),x=(3t^(2)+2)m,,(p),"Velocity of particle at t=1s is "8 m//s),((B),v=8t m//s,,(q),"Particle moves with uniform acceleration"),((C),a=16 t,,(r),"Particle moves with variable acceleration"),((D),v=6t-3t^(2),,(s),"Particle will change its direction some time"):}`

Explanation

To solve the problem, we need to match the equations of motion provided in Column I with the corresponding statements in Column II. Let's analyze each equation step by step.

### Step-by-Step Solution:

1. **Analyze Column I - Part A:**
- Given: \( x = 3t^2 + 2 \) m
- To find the velocity \( v \), we differentiate \( x \) with respect to \( t \):
\[ v = \frac{dx}{dt} = \frac{d}{dt}(3t^2 + 2) = 6t \text{ m/s} \]
- To find the acceleration \( a \), we differentiate \( v \) with respect to \( t \):
\[ a = \frac{dv}{dt} = \frac{d}{dt}(6t) = 6 \text{ m/s}^2 \]
- Since the acceleration is constant, the particle moves with uniform acceleration. Thus, **A matches with Q**.

2. **Analyze Column I - Part B:**
- Given: \( v = 8t \) m/s
- At \( t = 1 \) s:
\[ v = 8 \times 1 = 8 \text{ m/s} \]
- The acceleration \( a \) is:
\[ a = \frac{dv}{dt} = 8 \text{ m/s}^2 \]
- Since the acceleration is constant, this indicates uniform acceleration. Thus, **B matches with P and Q**.

3. **Analyze Column I - Part C:**
- Given: \( a = 16t \) m/s²
- Since acceleration depends on time, it is variable. Thus, **C matches with R**.

4. **Analyze Column I - Part D:**
- Given: \( v = 6t - 3t^2 \)
- To find acceleration:
\[ a = \frac{dv}{dt} = 6 - 6t \]
- The acceleration is variable since it depends on \( t \).
- To find when the particle changes direction, set \( v = 0 \):
\[ 6t - 3t^2 = 0 \implies 3t(2 - t) = 0 \implies t = 0 \text{ or } t = 2 \]
- The particle changes direction at \( t = 2 \) s. Thus, **D matches with S**.

### Final Matching:
- A matches with Q
- B matches with P and Q
- C matches with R
- D matches with S

### Summary of Matches:
- A → Q
- B → P, Q
- C → R
- D → S