Displacement of a particle executing SHM s `x= 10 ( cos pi t + sin pi t)`. Its maximum speed is

Options

A. `5 pi m//s`
B. ` 10 pi m//s`
C. ` 10 sqrt(2) pi m//s`
D. ` 5 sqrt(2 ) pi m//s`

Answer

To find the maximum speed of a particle executing simple harmonic motion (SHM) given the displacement equation \( x = 10 (\cos(\pi t) + \sin(\pi t)) \), we can follow these steps:

### Step 1: Rewrite the Displacement Equation
The given displacement equation is:
\[ x = 10 (\cos(\pi t) + \sin(\pi t)) \]
We can factor out \( \sqrt{2} \) to simplify the expression. We know that:
\[ \cos(\theta) + \sin(\theta) = \sqrt{2} \left( \cos(\theta - \frac{\pi}{4}) \right) \]
Thus, we can rewrite the displacement as:
\[ x = 10 \sqrt{2} \cos\left(\pi t - \frac{\pi}{4}\right) \]

### Step 2: Identify Amplitude and Angular Frequency
From the rewritten equation, we can identify:
- Amplitude \( A = 10\sqrt{2} \)
- Angular frequency \( \omega = \pi \)

### Step 3: Calculate Maximum Speed
The maximum speed \( v_{\text{max}} \) in SHM is given by the formula:
\[ v_{\text{max}} = A \omega \]
Substituting the values we found:
\[ v_{\text{max}} = (10\sqrt{2})(\pi) \]
Calculating this gives:
\[ v_{\text{max}} = 10\sqrt{2} \pi \text{ m/s} \]

### Final Answer
Thus, the maximum speed of the particle is:
\[ \boxed{10\sqrt{2} \pi \text{ m/s}} \]

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