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 step by step, we need to analyze the motion of the body along the x-axis given its velocity function and initial position.

### Step 1: Understand the velocity function
The velocity of the body is given by:
\[ v(t) = 5 \sin(2t) \, \text{m/s} \]

### Step 2: Determine the time intervals when the body crosses the origin
The body crosses the origin when its position \( x(t) \) becomes zero. We start with the initial position:
\[ x_i = -2.5 \, \text{m} \]

### Step 3: Find the expression for position
To find the position as a function of time, we need to integrate the velocity function:
\[ x(t) = x_i + \int v(t) \, dt \]
Substituting the velocity:
\[ x(t) = -2.5 + \int 5 \sin(2t) \, dt \]

### Step 4: Perform the integration
The integral of \( 5 \sin(2t) \) is:
\[ \int 5 \sin(2t) \, dt = -\frac{5}{2} \cos(2t) + C \]
To find the constant \( C \), we evaluate at \( t = 0 \):
\[ x(0) = -2.5 = -\frac{5}{2} \cos(0) + C \implies -2.5 = -\frac{5}{2} + C \implies C = -2.5 + 2.5 = 0 \]
Thus, the position function becomes:
\[ x(t) = -2.5 - \frac{5}{2} \cos(2t) \]

### Step 5: Set the position function to zero to find crossing times
To find when the body crosses the origin:
\[ 0 = -2.5 - \frac{5}{2} \cos(2t) \]
Rearranging gives:
\[ \frac{5}{2} \cos(2t) = -2.5 \implies \cos(2t) = -1 \]

### Step 6: Solve for \( t \)
The cosine function equals -1 at:
\[ 2t = (2n + 1) \pi \quad \text{for integers } n \]
Thus:
\[ t = \frac{(2n + 1) \pi}{2} \]
For \( n = 0, 1, 2, \ldots \), we find the times:
- For \( n = 0 \): \( t = \frac{\pi}{2} \)
- For \( n = 1 \): \( t = \frac{3\pi}{2} \)
- For \( n = 2 \): \( t = \frac{5\pi}{2} \)
- For \( n = 3 \): \( t = \frac{7\pi}{2} \)

### Step 7: Count the number of crossings in the interval [0, 2.5]
We need to check which of these times are less than or equal to 2.5 seconds:
- \( \frac{\pi}{2} \approx 1.57 \)
- \( \frac{3\pi}{2} \approx 4.71 \) (exceeds 2.5)
- \( \frac{5\pi}{2} \approx 7.85 \) (exceeds 2.5)
- \( \frac{7\pi}{2} \approx 11.00 \) (exceeds 2.5)

Thus, the body crosses the origin **twice** in the interval [0, 2.5].

### Step 8: Calculate the displacement in 2.5 seconds
The net displacement after 2.5 seconds can be calculated by evaluating \( x(2.5) \):
\[ x(2.5) = -2.5 - \frac{5}{2} \cos(5) \]
Using a calculator, find \( \cos(5) \) and substitute it in to find the exact displacement.

### Conclusion
1. The body crosses the origin **twice** in the interval [0, 2.5].
2. The displacement at \( t = 2.5 \) seconds can be calculated from the expression derived.