Two people start at the same place and walk around a circular lake in opposite directions. One has an angular speed of `1.7xx10^(-3)` rad/s while the other has an angular speed of `3.4xx10^(-3)` rad/s. How long will it be before they meet?

Options

A. 600 s
B. 2400 s
C. 1231 s
D. 1800 s

Answer

To solve the problem, we need to find out how long it will take for two people walking around a circular lake in opposite directions to meet. We know their angular speeds and can use the concept of relative angular velocity.

### Step-by-Step Solution:

1. **Identify the Angular Speeds**:
- Let the angular speed of person A be \( \omega_A = 1.7 \times 10^{-3} \) rad/s.
- Let the angular speed of person B be \( \omega_B = 3.4 \times 10^{-3} \) rad/s.

2. **Determine the Relative Angular Speed**:
- Since they are walking in opposite directions, we can add their angular speeds to find the relative angular speed:
\[ \omega_{relative} = \omega_A + \omega_B = 1.7 \times 10^{-3} + 3.4 \times 10^{-3} = 5.1 \times 10^{-3} \text{ rad/s} \]

3. **Calculate the Total Angular Displacement for Meeting**:
- For them to meet, the total angular displacement must equal \( 2\pi \) radians (a full circle).
\[ \theta = 2\pi \text{ radians} \]

4. **Use the Formula for Angular Displacement**:
- The relationship between angular displacement, angular speed, and time is given by:
\[ \theta = \omega_{relative} \times t \]
- Rearranging to solve for time \( t \):
\[ t = \frac{\theta}{\omega_{relative}} = \frac{2\pi}{5.1 \times 10^{-3}} \]

5. **Calculate the Time**:
- Plugging in the values:
\[ t = \frac{2\pi}{5.1 \times 10^{-3}} \approx \frac{6.2832}{5.1 \times 10^{-3}} \approx 1231.65 \text{ seconds} \]

6. **Final Answer**:
- Therefore, the time it will take for them to meet is approximately \( 1232 \) seconds.