A particle moves along the x - axis according to `x = A[1 + sin omega t]`. What distance does is travel in time interval from `t = 0` to `t = 2.5pi//omega` ?

Options

A. `4A`
B. `6 A`
C. `5 A`
D. `3 A`

Explanation

To solve the problem of finding the distance traveled by a particle moving along the x-axis according to the equation \( x = A[1 + \sin(\omega t)] \) from \( t = 0 \) to \( t = \frac{2.5\pi}{\omega} \), we can follow these steps:

### Step 1: Identify the Time Period
The time period \( T \) of the motion can be derived from the angular frequency \( \omega \):
\[ T = \frac{2\pi}{\omega} \]

### Step 2: Break Down the Time Interval
The total time interval from \( t = 0 \) to \( t = \frac{2.5\pi}{\omega} \) can be expressed as:
\[ \frac{2.5\pi}{\omega} = \frac{2\pi}{\omega} + \frac{0.5\pi}{\omega} \]
This indicates that the particle first completes one full oscillation (time period \( T \)) and then moves for an additional \( \frac{0.5\pi}{\omega} \).

### Step 3: Calculate Distance for One Full Oscillation
In one complete oscillation (from \( t = 0 \) to \( t = T \)), the particle moves from its starting position to the positive extreme, back to the mean position, to the negative extreme, and back to the mean position. The total distance covered in one full oscillation is:
\[ \text{Distance in one period} = A + A + A + A = 4A \]

### Step 4: Calculate Distance for the Additional Time Interval
Next, we need to find the distance traveled during the additional time interval of \( \frac{0.5\pi}{\omega} \). This time interval is \( \frac{T}{4} \) (since \( T = \frac{2\pi}{\omega} \)):
\[ \frac{0.5\pi}{\omega} = \frac{T}{4} \]
During this time, the particle moves from the mean position to the positive extreme, which covers a distance of:
\[ \text{Distance in } \frac{T}{4} = A \]

### Step 5: Total Distance Traveled
Now, we can sum the distances from both parts:
\[ \text{Total Distance} = \text{Distance in one period} + \text{Distance in } \frac{T}{4} = 4A + A = 5A \]

### Conclusion
Thus, the total distance traveled by the particle from \( t = 0 \) to \( t = \frac{2.5\pi}{\omega} \) is:
\[ \boxed{5A} \]