body moves along x-axis with velocity v = 5 sin(2t)m/s starting from ` x_i =-2.5 m`
Which of the following statements are correct ?

Options

A. Body crosses the origin twice in ` 2 pi sec`
B. Body crosses the origin 4 times in ` 2pi sec`
C. Total displacement of body in ` 2pi sec "is" 5 m`
D. Total displacement of body in ` 2pi sec ` is 0

Explanation

To solve the problem, we need to analyze the motion of a body moving along the x-axis with a given velocity function and determine the correct statements regarding its displacement and whether it crosses the origin.

### Step-by-Step Solution:

1. **Identify the Given Information**:
- The velocity of the body is given by \( v(t) = 5 \sin(2t) \) m/s.
- The initial position is \( x_i = -2.5 \) m.

2. **Determine the Displacement**:
- Displacement \( x(t) \) can be found by integrating the velocity function:
\[ x(t) = x_i + \int_0^t v(t) \, dt \]
- Here, we need to find the displacement from \( t = 0 \) to \( t = 2\pi \):
\[ x(t) = -2.5 + \int_0^{2\pi} 5 \sin(2t) \, dt \]

3. **Calculate the Integral**:
- The integral of \( 5 \sin(2t) \) can be calculated as follows:
\[ \int 5 \sin(2t) \, dt = -\frac{5}{2} \cos(2t) + C \]
- Evaluating the definite integral from \( 0 \) to \( 2\pi \):
\[ \int_0^{2\pi} 5 \sin(2t) \, dt = \left[-\frac{5}{2} \cos(2t)\right]_0^{2\pi} \]
- Substitute the limits:
\[ = -\frac{5}{2} (\cos(4\pi) - \cos(0)) = -\frac{5}{2} (1 - 1) = 0 \]

4. **Final Displacement Calculation**:
- Thus, the total displacement after \( 2\pi \) seconds is:
\[ x(2\pi) = -2.5 + 0 = -2.5 \, \text{m} \]

5. **Determine if the Particle Crosses the Origin**:
- The displacement function can be expressed as:
\[ x(t) = -2.5 - \frac{5}{2} \cos(2t) \]
- To find when it crosses the origin, set \( x(t) = 0 \):
\[ 0 = -2.5 - \frac{5}{2} \cos(2t) \]
\[ \frac{5}{2} \cos(2t) = -2.5 \implies \cos(2t) = -1 \]
- The cosine function equals -1 at \( 2t = \pi + 2n\pi \) for \( n \in \mathbb{Z} \). Thus:
\[ t = \frac{\pi}{2} + n\pi \]
- The first occurrence is at \( t = \frac{\pi}{2} \), which is within the interval \( [0, 2\pi] \).

### Conclusion:
- The body does cross the origin at \( t = \frac{\pi}{2} \).
- The total displacement after \( 2\pi \) seconds is \( -2.5 \, \text{m} \).