To solve the problem, we need to find the total distance traveled by a body moving along the x-axis with a given velocity function \( v(t) = 5 \sin(2t) \) over a time interval of \( \frac{3\pi}{2} \) seconds, starting from an initial position of \( x_i = -2.5 \, \text{m} \).
### Step-by-Step Solution:
1. **Understand the Velocity Function**:
The velocity of the body is given by \( v(t) = 5 \sin(2t) \). This function oscillates between -5 m/s and 5 m/s.
2. **Determine the Time Interval**:
We are interested in the time interval from \( t = 0 \) to \( t = \frac{3\pi}{2} \).
3. **Find the Points Where Velocity is Zero**:
To find the total distance traveled, we need to determine when the velocity changes direction (i.e., when \( v(t) = 0 \)):
\[
5 \sin(2t) = 0 \implies \sin(2t) = 0 \implies 2t = n\pi \implies t = \frac{n\pi}{2}
\]
For \( n = 0, 1, 2, 3 \):
- \( t = 0 \)
- \( t = \frac{\pi}{2} \)
- \( t = \pi \)
- \( t = \frac{3\pi}{2} \)
4. **Calculate the Total Distance**:
We will calculate the distance traveled in each segment:
- From \( t = 0 \) to \( t = \frac{\pi}{2} \):
\[
\text{Distance} = \int_0^{\frac{\pi}{2}} 5 \sin(2t) \, dt
\]
\[
= -\frac{5}{2} \cos(2t) \bigg|_0^{\frac{\pi}{2}} = -\frac{5}{2} \left( \cos(\pi) - \cos(0) \right) = -\frac{5}{2} \left( -1 - 1 \right) = 5 \, \text{m}
\]
- From \( t = \frac{\pi}{2} \) to \( t = \pi \):
The velocity is negative in this interval, so the distance will still be positive:
\[
\text{Distance} = \int_{\frac{\pi}{2}}^{\pi} 5 \sin(2t) \, dt = -\frac{5}{2} \cos(2t) \bigg|_{\frac{\pi}{2}}^{\pi} = -\frac{5}{2} \left( \cos(2\pi) - \cos(\pi) \right) = -\frac{5}{2} \left( 1 - (-1) \right) = -\frac{5}{2} \cdot 2 = -5 \, \text{m}
\]
Since the distance is positive, we take it as \( 5 \, \text{m} \).
- From \( t = \pi \) to \( t = \frac{3\pi}{2} \):
The velocity is again positive:
\[
\text{Distance} = \int_{\pi}^{\frac{3\pi}{2}} 5 \sin(2t) \, dt = -\frac{5}{2} \cos(2t) \bigg|_{\pi}^{\frac{3\pi}{2}} = -\frac{5}{2} \left( \cos(3\pi) - \cos(2\pi) \right) = -\frac{5}{2} \left( -1 - 1 \right) = 5 \, \text{m}
\]
5. **Total Distance**:
Adding all the distances:
\[
\text{Total Distance} = 5 \, \text{m} + 5 \, \text{m} + 5 \, \text{m} = 15 \, \text{m}
\]
6. **Final Position**:
The initial position is \( x_i = -2.5 \, \text{m} \). The final position after traveling 15 m is:
\[
x_f = x_i + \text{Total Distance} = -2.5 \, \text{m} + 15 \, \text{m} = 12.5 \, \text{m}
\]
### Conclusion:
The total distance traveled by the body in \( \frac{3\pi}{2} \) seconds is **15 meters**.
