The equation of a travelling wave is
`y(x, t) = 0.02 sin ((x)/(0.05) + (t)/(0.01)) m`
Find (a) the wave velocity and
(b) the particle velocity at `x = 0.2 m and t = 0.3 s`.

Explanation

To solve the problem, we will break it down into two parts: finding the wave velocity and finding the particle velocity.

### Part (a): Finding the Wave Velocity

1. **Identify the wave equation**: The given wave equation is:
\[ y(x, t) = 0.02 \sin\left(\frac{x}{0.05} + \frac{t}{0.01}\right) \text{ m} \]

2. **Identify the wave parameters**: From the wave equation, we can identify:
- The angular frequency \(\omega\) is given by the coefficient of \(t\):
\[ \omega = \frac{1}{0.01} = 100 \text{ rad/s} \]
- The wave number \(k\) is given by the coefficient of \(x\):
\[ k = \frac{1}{0.05} = 20 \text{ rad/m} \]

3. **Calculate the wave velocity**: The wave velocity \(v\) can be calculated using the formula:
\[ v = \frac{\omega}{k} \]
Substituting the values of \(\omega\) and \(k\):
\[ v = \frac{100}{20} = 5 \text{ m/s} \]

### Part (b): Finding the Particle Velocity

1. **Find the partial derivative of \(y\) with respect to \(t\)**: The particle velocity \(v_p\) is given by:
\[ v_p = \frac{\partial y}{\partial t} \]
We differentiate \(y(x, t)\):
\[ \frac{\partial y}{\partial t} = 0.02 \cdot \cos\left(\frac{x}{0.05} + \frac{t}{0.01}\right) \cdot \frac{1}{0.01} \]

2. **Substitute \(x = 0.2 \, \text{m}\) and \(t = 0.3 \, \text{s}\)**: First, we calculate the argument of the cosine:
\[ \frac{0.2}{0.05} + \frac{0.3}{0.01} = 4 + 30 = 34 \]
Therefore, we have:
\[ \frac{\partial y}{\partial t} = 0.02 \cdot \cos(34) \cdot 100 \]
Simplifying this gives:
\[ \frac{\partial y}{\partial t} = 2 \cdot \cos(34) \]

3. **Calculate the numerical value**: Now we need to calculate \(2 \cdot \cos(34)\). Using a calculator:
\[ \cos(34^\circ) \approx 0.829 \]
Thus:
\[ v_p \approx 2 \cdot 0.829 \approx 1.658 \text{ m/s} \]

### Final Answers
- (a) The wave velocity \(v\) is **5 m/s**.
- (b) The particle velocity \(v_p\) at \(x = 0.2 \, \text{m}\) and \(t = 0.3 \, \text{s}\) is approximately **1.658 m/s**.