The displacement of a particle executing S.H.M. is given by `y = 10 sin [6t + (pi)/(3)]` where y is in metres and t is in seconds. Then the initial displacement and velocity of the particle is

Options

A. `5sqrt3 m and 30 ms^(-1)`
B. `15 m and 5 sqrt3 ms^(-1)`
C. `15 sqrt3 and 30 ms^(-1)`
D. `20 sqrt3 and 30 ms^(-1)`

Explanation

To find the initial displacement and velocity of the particle executing Simple Harmonic Motion (S.H.M.) given by the equation:

\[ y = 10 \sin(6t + \frac{\pi}{3}) \]

we will follow these steps:

### Step 1: Find the Initial Displacement
The initial displacement occurs at \( t = 0 \). We can substitute \( t = 0 \) into the displacement equation.

\[ y(0) = 10 \sin(6 \cdot 0 + \frac{\pi}{3}) \]

This simplifies to:

\[ y(0) = 10 \sin(\frac{\pi}{3}) \]

### Step 2: Calculate \( \sin(\frac{\pi}{3}) \)
The value of \( \sin(\frac{\pi}{3}) \) is known to be \( \frac{\sqrt{3}}{2} \).

\[ y(0) = 10 \cdot \frac{\sqrt{3}}{2} \]

### Step 3: Simplify the Initial Displacement
Now, we can simplify this expression:

\[ y(0) = 5\sqrt{3} \text{ meters} \]

### Step 4: Find the Initial Velocity
The velocity in S.H.M. can be found by differentiating the displacement with respect to time \( t \).

\[ v(t) = \frac{dy}{dt} = \frac{d}{dt} [10 \sin(6t + \frac{\pi}{3})] \]

Using the chain rule, we have:

\[ v(t) = 10 \cdot 6 \cos(6t + \frac{\pi}{3}) = 60 \cos(6t + \frac{\pi}{3}) \]

### Step 5: Calculate the Initial Velocity
Now, we find the initial velocity at \( t = 0 \):

\[ v(0) = 60 \cos(6 \cdot 0 + \frac{\pi}{3}) \]

This simplifies to:

\[ v(0) = 60 \cos(\frac{\pi}{3}) \]

### Step 6: Calculate \( \cos(\frac{\pi}{3}) \)
The value of \( \cos(\frac{\pi}{3}) \) is known to be \( \frac{1}{2} \).

\[ v(0) = 60 \cdot \frac{1}{2} \]

### Step 7: Simplify the Initial Velocity
Now, we can simplify this expression:

\[ v(0) = 30 \text{ meters per second} \]

### Final Results
Thus, the initial displacement and velocity of the particle are:

- Initial Displacement: \( 5\sqrt{3} \) meters
- Initial Velocity: \( 30 \) meters per second

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