The position of a paricle moving along `x-`axis depends on time as `x = 2(1 - e^(-31))`. Choose the correct statement(s).

Options

A. Total displacement of the particle is `2` units
B. Total distance covered by particle is `2` units.
C. Velocity of the particle will change its direction at some instant.
D. Velocity of particle will not change its direction throughout the motion.

Explanation

To solve the problem, we need to analyze the position function given for the particle moving along the x-axis, which is:

\[ x(t) = 2(1 - e^{-3t}) \]

### Step 1: Determine the initial position at \( t = 0 \)

Substituting \( t = 0 \) into the position equation:

\[ x(0) = 2(1 - e^{-3 \cdot 0}) = 2(1 - e^{0}) = 2(1 - 1) = 2 \cdot 0 = 0 \]

**Hint:** The exponential function \( e^0 \) equals 1, which simplifies the calculation.

### Step 2: Determine the position as \( t \to \infty \)

Now, we need to find the position of the particle as time approaches infinity:

\[ x(\infty) = 2(1 - e^{-3 \cdot \infty}) = 2(1 - 0) = 2 \]

**Hint:** As \( t \) increases, \( e^{-3t} \) approaches 0, simplifying the expression.

### Step 3: Calculate the velocity

The velocity \( v(t) \) is the derivative of the position function with respect to time:

\[ v(t) = \frac{dx}{dt} = \frac{d}{dt}[2(1 - e^{-3t})] \]

Using the chain rule:

\[ v(t) = 2 \cdot 0 - 2 \cdot (-3)e^{-3t} = 6e^{-3t} \]

**Hint:** Remember that the derivative of \( e^{kt} \) is \( ke^{kt} \).

### Step 4: Analyze the velocity

Since \( e^{-3t} \) is always positive for all \( t \geq 0 \), it follows that:

\[ v(t) = 6e^{-3t} > 0 \quad \text{for all } t \geq 0 \]

**Hint:** The exponential function is always positive, which indicates that the velocity does not change sign.

### Step 5: Conclusion about displacement and distance

1. **Total Displacement:** The total displacement from \( t = 0 \) to \( t = \infty \) is \( x(\infty) - x(0) = 2 - 0 = 2 \) units.

2. **Distance Covered:** Since the velocity is always positive, the distance covered is equal to the total displacement, which is also 2 units.

3. **Direction of Velocity:** Since the velocity is always positive, it does not change direction.

### Final Statements

- The total displacement of the particle is 2 units.
- The distance covered by the particle is also 2 units.
- The velocity of the particle does not change direction at any instant.

**Correct Statements:**
- The total displacement of the particle is 2 units.
- The distance covered by the particle is 2 units.
- The velocity of the particle does not change direction.