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 analyze the motion of the particle described by the equation \( x = a \cos(\pi t) \).

### Step-by-Step Solution:

1. **Identify the Motion Type**:
The equation given is \( x = a \cos(\pi t) \). This is the equation of Simple Harmonic Motion (SHM), where \( A = a \) is the amplitude and \( \omega = \pi \) is the angular frequency.

2. **Determine the Time Period**:
The time period \( T \) of SHM is given by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting \( \omega = \pi \):
\[ T = \frac{2\pi}{\pi} = 2 \text{ seconds} \]

3. **Calculate the Distance Covered in One Complete Cycle**:
In one complete cycle (which takes \( T = 2 \) seconds), the particle moves from \( +a \) to \( -a \) and back to \( +a \). The total distance covered in one complete cycle is:
\[ \text{Distance} = a + a + a = 4a \]

4. **Determine the Position at \( t = 2.5 \) seconds**:
Since \( 2.5 \) seconds is \( 0.5 \) seconds more than \( 2 \) seconds, we need to find out how far the particle moves in the additional \( 0.5 \) seconds after completing one full cycle.

5. **Calculate the Position at \( t = 0.5 \) seconds**:
To find the position at \( t = 2.5 \) seconds, we first find the position at \( t = 0.5 \) seconds:
\[ x(0.5) = a \cos(\pi \times 0.5) = a \cos\left(\frac{\pi}{2}\right) = a \times 0 = 0 \]
Thus, at \( t = 0.5 \) seconds, the particle is at the mean position (0).

6. **Calculate the Total Distance Covered**:
The total distance covered by the particle in \( 2.5 \) seconds is the distance covered in the first \( 2 \) seconds plus the distance covered in the next \( 0.5 \) seconds:
- Distance covered in \( 2 \) seconds: \( 4a \)
- Distance covered in \( 0.5 \) seconds: from \( 0 \) to \( +a \) (since it moves from the mean position to the maximum amplitude):
- This distance is \( a \).

Therefore, the total distance covered in \( 2.5 \) seconds is:
\[ \text{Total Distance} = 4a + a = 5a \]

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

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