Find the distance covered by a particle from time `t = 0` to `t = 6` sec when the particle follow the movament according to `y = a cos ((pi)/(4))t`

Options

A. a
B. `2a`
C. `3a`
D. `4a`

Explanation

To find the distance covered by the particle from time \( t = 0 \) to \( t = 6 \) seconds, we can follow these steps:

### Step 1: Identify the equation of motion
The motion of the particle is given by the equation:
\[ y = a \cos\left(\frac{\pi}{4} t\right) \]
where \( a \) is the amplitude of the motion.

### Step 2: Determine the angular frequency and the time period
From the equation, we can identify the angular frequency \( \omega \):
\[ \omega = \frac{\pi}{4} \]
The time period \( T \) can be calculated using the formula:
\[ T = \frac{2\pi}{\omega} = \frac{2\pi}{\frac{\pi}{4}} = 8 \text{ seconds} \]

### Step 3: Understand the motion over one complete cycle
In one complete cycle (8 seconds), the particle moves from the maximum displacement \( a \) to \( -a \) and back to \( a \). The total distance covered in one complete cycle is:
\[ \text{Distance in one cycle} = 4a \]
This is because the particle travels from \( a \) to \( 0 \) (a distance of \( a \)), then from \( 0 \) to \( -a \) (a distance of \( a \)), then from \( -a \) back to \( 0 \) (another distance of \( a \)), and finally from \( 0 \) back to \( a \) (another distance of \( a \)).

### Step 4: Calculate the distance covered in 6 seconds
Since the total distance covered in 8 seconds is \( 4a \), we can find the distance covered in 1 second:
\[ \text{Distance per second} = \frac{4a}{8} = \frac{a}{2} \]
Now, to find the distance covered in 6 seconds:
\[ \text{Distance in 6 seconds} = 6 \times \frac{a}{2} = 3a \]

### Final Answer
The distance covered by the particle from \( t = 0 \) to \( t = 6 \) seconds is:
\[ \boxed{3a} \]