A particle moves according to the equation `x= a cos pi t`. The distance covered by it in `2.5` s is

Options

A. 3a
B. 5a
C. 2a
D. 9a

Answer

To solve the problem, we need to determine the distance covered by a particle moving according to the equation \( x = a \cos(\pi t) \) in a time interval of \( 2.5 \) seconds.

### Step-by-Step Solution:

1. **Identify the Motion Parameters**:
The equation of motion is given as \( x = a \cos(\pi t) \). Here, \( a \) is the amplitude, and \( \omega = \pi \) is the angular frequency.

2. **Determine the Time Period**:
The time period \( T \) can be calculated using the formula:
\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{\pi} = 2 \text{ seconds} \]

3. **Determine the Number of Cycles in 2.5 seconds**:
In \( 2.5 \) seconds, the number of complete cycles \( n \) can be calculated as:
\[ n = \frac{2.5}{T} = \frac{2.5}{2} = 1.25 \text{ cycles} \]

4. **Analyze the Motion**:
- In one complete cycle (2 seconds), the particle moves from \( +a \) to \( -a \) and back to \( +a \). The total distance covered in one complete cycle is \( 4a \) (from \( +a \) to \( 0 \) to \( -a \) to \( 0 \) and back to \( +a \)).
- In the first complete cycle (0 to 2 seconds), the particle covers a distance of \( 4a \).

5. **Calculate the Distance Covered in the Remaining 0.5 Seconds**:
- After 2 seconds, the particle starts its next cycle. At \( t = 2.5 \) seconds, we need to find the position:
\[ x = a \cos(\pi \cdot 2.5) = a \cos(2.5\pi) \]
- Since \( \cos(2.5\pi) = -1 \), the position at \( t = 2.5 \) seconds is \( x = -a \).

6. **Distance Covered in the Last 0.5 Seconds**:
- From \( t = 2 \) seconds to \( t = 2.5 \) seconds, the particle moves from \( 0 \) (mean position) to \( -a \) (extreme position).
- The distance covered during this time is \( a \).

7. **Total Distance Covered**:
- The total distance covered in \( 2.5 \) seconds is:
\[ \text{Total Distance} = \text{Distance in 1st cycle} + \text{Distance in last 0.5 seconds} = 4a + a = 5a \]

### Final Answer:
The distance covered by the particle in \( 2.5 \) seconds is \( 5a \).

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