The displacement of a particle executing `S.H.M` from its mean position is given by `x = 0.5 sin (10 pit +phi) cos (10pit +phi)`. The ratio of the maximum velocity to the maximum acceleration of the body is given by

Options

A. `20pi`
B. `(1)/(20pi)`
C. `(1)/(10pi)`
D. `10pi`

Explanation

To solve the problem, we need to find the ratio of the maximum velocity to the maximum acceleration of a particle executing simple harmonic motion (S.H.M.) given the displacement equation:

\[ x = 0.5 \sin(10 \pi t + \phi) \cos(10 \pi t + \phi) \]

### Step 1: Simplify the Displacement Equation
We can use the trigonometric identity:

\[ \sin A \cos A = \frac{1}{2} \sin(2A) \]

Applying this identity, we rewrite the displacement:

\[ x = 0.5 \cdot \sin(10 \pi t + \phi) \cdot \cos(10 \pi t + \phi) = 0.5 \cdot \frac{1}{2} \sin(2(10 \pi t + \phi)) = 0.25 \sin(20 \pi t + 2\phi) \]

### Step 2: Find the Maximum Velocity
The velocity \( v \) is the time derivative of the displacement \( x \):

\[ v = \frac{dx}{dt} = \frac{d}{dt}(0.25 \sin(20 \pi t + 2\phi)) \]

Using the derivative of sine:

\[ v = 0.25 \cdot 20 \pi \cos(20 \pi t + 2\phi) = 5 \pi \cos(20 \pi t + 2\phi) \]

The maximum velocity \( v_0 \) occurs when \( \cos(20 \pi t + 2\phi) = 1 \):

\[ v_0 = 5 \pi \]

### Step 3: Find the Maximum Acceleration
The acceleration \( a \) is the time derivative of the velocity \( v \):

\[ a = \frac{dv}{dt} = \frac{d}{dt}(5 \pi \cos(20 \pi t + 2\phi)) \]

Using the derivative of cosine:

\[ a = -5 \pi \cdot 20 \pi \sin(20 \pi t + 2\phi) = -100 \pi^2 \sin(20 \pi t + 2\phi) \]

The maximum acceleration \( a_0 \) occurs when \( \sin(20 \pi t + 2\phi) = 1 \):

\[ a_0 = 100 \pi^2 \]

### Step 4: Calculate the Ratio of Maximum Velocity to Maximum Acceleration
Now we can find the ratio of maximum velocity to maximum acceleration:

\[ \text{Ratio} = \frac{v_0}{a_0} = \frac{5 \pi}{100 \pi^2} = \frac{5}{100 \pi} = \frac{1}{20 \pi} \]

Thus, the ratio of the maximum velocity to the maximum acceleration is:

\[ \frac{1}{20 \pi} \]

### Final Answer
The ratio of the maximum velocity to the maximum acceleration of the body is \( \frac{1}{20 \pi} \).

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