body moves along x-axis with velocity v = 5 sin(2t)m/s starting from ` x_i =-2.5 m`Ratio of displacement to distance traveled by body from ` t= 0s "to" t= 3pi //2 sec ` is :

Options

A. `1:1`
B. 0
C. `1:3`
D. `1:2`

Explanation

To solve the problem step by step, we will follow these instructions:

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

### Step 2: Relate velocity to displacement
We know that velocity is the derivative of displacement with respect to time:
\[ v = \frac{dx}{dt} \]
This implies:
\[ dx = v \, dt \]
Substituting the velocity function:
\[ dx = 5 \sin(2t) \, dt \]

### Step 3: Integrate to find displacement
To find the displacement \( x \), we integrate the equation from \( t = 0 \) to \( t = T \):
\[ x = \int_0^T 5 \sin(2t) \, dt \]
Calculating the integral:
\[ x = 5 \left[-\frac{1}{2} \cos(2t)\right]_0^T \]
\[ x = -\frac{5}{2} \left[\cos(2T) - \cos(0)\right] \]
\[ x = -\frac{5}{2} \left[\cos(2T) - 1\right] \]

### Step 4: Substitute limits for \( T = \frac{3\pi}{2} \)
Now, we substitute \( T = \frac{3\pi}{2} \):
\[ x = -\frac{5}{2} \left[\cos(3\pi) - 1\right] \]
Since \( \cos(3\pi) = -1 \):
\[ x = -\frac{5}{2} \left[-1 - 1\right] \]
\[ x = -\frac{5}{2} \left[-2\right] \]
\[ x = 5 \, \text{m} \]

### Step 5: Calculate the total distance traveled
The body moves from \( x_i = -2.5 \, \text{m} \) to \( x_f = 5 \, \text{m} \). The total distance traveled is the sum of the distances in each segment of the motion:
1. From \( -2.5 \) to \( 5 \) (forward): \( 5 - (-2.5) = 7.5 \, \text{m} \)
2. The body also returns back to \( -2.5 \) before reaching \( 5 \) again, which adds another \( 7.5 \, \text{m} \).

Thus, the total distance traveled is:
\[ \text{Total Distance} = 7.5 + 7.5 = 15 \, \text{m} \]

### Step 6: Calculate the ratio of displacement to distance
The displacement is \( 5 \, \text{m} \) and the total distance traveled is \( 15 \, \text{m} \). Therefore, the ratio of displacement to distance is:
\[ \text{Ratio} = \frac{\text{Displacement}}{\text{Distance}} = \frac{5}{15} = \frac{1}{3} \]

### Final Answer
The ratio of displacement to distance traveled by the body from \( t = 0 \) to \( t = \frac{3\pi}{2} \) seconds is:
\[ \frac{1}{3} \]

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