A particle of mass 0.10 kg has its velocity varying according to the relation`v=pi sin (pi t+(pi)/(4))m//sec`
What is the maximum kinetic energy (in joules)?`(Take pi^(2)=10)`

Explanation

To find the maximum kinetic energy of the particle, we can follow these steps:

### Step 1: Identify the given parameters
The mass of the particle \( m = 0.10 \, \text{kg} \) and the velocity is given by the equation:
\[ v = \pi \sin\left(\pi t + \frac{\pi}{4}\right) \, \text{m/s} \]

### Step 2: Determine the maximum velocity
The maximum value of the sine function is 1. Therefore, the maximum velocity \( v_{\text{max}} \) can be calculated as:
\[ v_{\text{max}} = \pi \cdot 1 = \pi \, \text{m/s} \]

### Step 3: Calculate the maximum kinetic energy
The formula for kinetic energy \( KE \) is given by:
\[ KE = \frac{1}{2} m v^2 \]
To find the maximum kinetic energy, we substitute \( v_{\text{max}} \) into the kinetic energy formula:
\[ KE_{\text{max}} = \frac{1}{2} m v_{\text{max}}^2 = \frac{1}{2} m (\pi)^2 \]

### Step 4: Substitute the values
We know that \( \pi^2 = 10 \) (as given in the problem). Thus, we can substitute the values:
\[ KE_{\text{max}} = \frac{1}{2} \cdot 0.10 \cdot 10 \]

### Step 5: Perform the calculation
Calculating the above expression:
\[ KE_{\text{max}} = \frac{1}{2} \cdot 0.10 \cdot 10 = 0.5 \, \text{Joules} \]

### Final Answer
The maximum kinetic energy of the particle is:
\[ \boxed{0.5 \, \text{Joules}} \]

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