A particle is executing SHM about y =0 along y-axis. Its position at an instant is given by `y = (7m) sin (pit)`. Its average velocity for a time interval 0 to 0.5 s is

Options

A. 14m/s
B. 7m/s
C. `(1)/(7)`m/s
D. 28m/s

Answer

To find the average velocity of a particle executing simple harmonic motion (SHM) described by the equation \( y = 7 \sin(\pi t) \) over the time interval from \( t = 0 \) to \( t = 0.5 \) seconds, we will follow these steps:

### Step 1: Identify the Displacement at \( t = 0 \)
At \( t = 0 \):
\[ y_1 = 7 \sin(\pi \cdot 0) = 7 \sin(0) = 0 \, \text{m} \]

### Step 2: Identify the Displacement at \( t = 0.5 \)
At \( t = 0.5 \):
\[ y_2 = 7 \sin(\pi \cdot 0.5) = 7 \sin\left(\frac{\pi}{2}\right) = 7 \cdot 1 = 7 \, \text{m} \]

### Step 3: Calculate the Total Displacement
The total displacement (\( \Delta y \)) over the time interval from \( t = 0 \) to \( t = 0.5 \) seconds is:
\[ \Delta y = y_2 - y_1 = 7 \, \text{m} - 0 \, \text{m} = 7 \, \text{m} \]

### Step 4: Calculate the Total Time
The total time (\( \Delta t \)) for the interval is:
\[ \Delta t = t_2 - t_1 = 0.5 \, \text{s} - 0 \, \text{s} = 0.5 \, \text{s} \]

### Step 5: Calculate the Average Velocity
The average velocity (\( v_{\text{avg}} \)) is given by the formula:
\[ v_{\text{avg}} = \frac{\Delta y}{\Delta t} \]
Substituting the values:
\[ v_{\text{avg}} = \frac{7 \, \text{m}}{0.5 \, \text{s}} = 14 \, \text{m/s} \]

### Final Answer
The average velocity of the particle over the time interval from \( t = 0 \) to \( t = 0.5 \) seconds is:
\[ \boxed{14 \, \text{m/s}} \]