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: Understand the Kinetic Energy Formula
The kinetic energy (KE) of a particle is given by the formula:
\[ KE = \frac{1}{2} m v^2 \]
where \( m \) is the mass of the particle and \( v \) is its velocity.

### Step 2: Identify the Mass of the Particle
From the problem, we know that the mass \( m \) of the particle is:
\[ m = 0.10 \, \text{kg} \]

### Step 3: Determine the Velocity Function
The velocity of the particle is given by:
\[ v = \pi \sin\left(\pi t + \frac{\pi}{4}\right) \, \text{m/s} \]

### Step 4: Find the Maximum Velocity
The maximum value of the sine function is 1. Therefore, the maximum velocity \( v_{\text{max}} \) occurs when:
\[ \sin\left(\pi t + \frac{\pi}{4}\right) = 1 \]
This gives:
\[ v_{\text{max}} = \pi \times 1 = \pi \, \text{m/s} \]

### Step 5: Substitute Maximum Velocity into Kinetic Energy Formula
Now, we can substitute \( v_{\text{max}} \) into the kinetic energy formula:
\[ KE_{\text{max}} = \frac{1}{2} m v_{\text{max}}^2 = \frac{1}{2} \times 0.10 \times (\pi)^2 \]

### Step 6: Substitute the Value of \( \pi^2 \)
According to the problem, we take \( \pi^2 = 10 \):
\[ KE_{\text{max}} = \frac{1}{2} \times 0.10 \times 10 \]

### Step 7: Calculate the Maximum Kinetic Energy
Now, performing the calculation:
\[ KE_{\text{max}} = \frac{1}{2} \times 0.10 \times 10 = 0.05 \times 10 = 0.5 \, \text{J} \]

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

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