A particle moves clockwise in acircle of radius 1m with centre at `(x, y)=(1m, 0)`. It starts at rest at the origin at time `t=0`. Its speed increases at the constant rate of `(pi/2) m//s^(2)`. (i) Hpw long does it takes to travel halfway around the circle? (ii) What is the speed at that time?

Explanation

To solve the problem, we need to find out how long it takes for a particle to travel halfway around a circle of radius 1 meter, and what its speed is at that time. The particle starts at rest at the origin and accelerates at a constant rate of \(\frac{\pi}{2} \, \text{m/s}^2\).

### Step-by-Step Solution

1. **Determine the distance to travel halfway around the circle:**
- The circumference \(C\) of a circle is given by the formula:
\[ C = 2\pi r \]
- For a circle with a radius \(r = 1 \, \text{m}\):
\[ C = 2\pi \times 1 = 2\pi \, \text{m} \]
- Halfway around the circle is:
\[ s = \frac{C}{2} = \frac{2\pi}{2} = \pi \, \text{m} \]

2. **Use the equation of motion to find the time taken to travel this distance:**
- The equation of motion that relates distance \(s\), initial velocity \(u\), acceleration \(a\), and time \(t\) is:
\[ s = ut + \frac{1}{2} a t^2 \]
- Here, the initial velocity \(u = 0\) (since the particle starts at rest), and the acceleration \(a = \frac{\pi}{2} \, \text{m/s}^2\).
- Substituting the values into the equation:
\[ \pi = 0 \cdot t + \frac{1}{2} \cdot \frac{\pi}{2} \cdot t^2 \]
- This simplifies to:
\[ \pi = \frac{\pi}{4} t^2 \]
- To eliminate \(\pi\) from both sides (assuming \(\pi \neq 0\)):
\[ 1 = \frac{1}{4} t^2 \]
- Multiplying both sides by 4 gives:
\[ t^2 = 4 \]
- Taking the square root of both sides:
\[ t = 2 \, \text{s} \]

3. **Calculate the speed at that time:**
- The speed \(v\) of the particle at time \(t\) can be calculated using the formula:
\[ v = u + at \]
- Substituting the known values:
\[ v = 0 + \frac{\pi}{2} \cdot 2 \]
- This simplifies to:
\[ v = \pi \, \text{m/s} \]

### Final Answers
(i) The time taken to travel halfway around the circle is \(2 \, \text{s}\).

(ii) The speed at that time is \(\pi \, \text{m/s}\).