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![### Radar Resolution Problem
A circular radar antenna on a Coast Guard ship has a diameter of 2.10 meters and radiates at a frequency of 14.0 GHz. Two small boats are located 8.00 km away from the ship. How close together could the boats be and still be detected as two objects?
**Solution Area:**
\[ \text{Minimum detectable separation:} \ \_\_\_\_ \text{ meters} \]
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**Explanation:**
In this problem, you need to determine the minimum distance at which two boats can be detected as separate objects by the radar. This involves understanding the concepts of radar resolution and the Rayleigh criterion.
The Rayleigh criterion for the minimum angular resolution \(\theta\) can be given by:
\[ \theta = 1.22 \frac{\lambda}{D} \]
where,
- \(\lambda\) is the wavelength of the radar frequency,
- \(D\) is the diameter of the radar antenna.
The wavelength \(\lambda\) can be calculated using the speed of light \(c\) and the frequency \(f\):
\[ \lambda = \frac{c}{f} \]
Given:
- Diameter of radar antenna, \(D = 2.10 \ \text{m} \)
- Frequency, \(f = 14.0 \ \text{GHz} \)
- Distance to the boats, \(d = 8.00 \ \text{km} \)
Using these formulas, you can find the minimum detectable separation \(d_{\text{min}}\).
Feel free to dive into the "Read It" or "Watch It" sections for more detailed guidance!
---
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Transcribed Image Text:### Radar Resolution Problem
A circular radar antenna on a Coast Guard ship has a diameter of 2.10 meters and radiates at a frequency of 14.0 GHz. Two small boats are located 8.00 km away from the ship. How close together could the boats be and still be detected as two objects?
**Solution Area:**
\[ \text{Minimum detectable separation:} \ \_\_\_\_ \text{ meters} \]
**Need Help?**
- [Read It](#)
- [Watch It](#)
**Explanation:**
In this problem, you need to determine the minimum distance at which two boats can be detected as separate objects by the radar. This involves understanding the concepts of radar resolution and the Rayleigh criterion.
The Rayleigh criterion for the minimum angular resolution \(\theta\) can be given by:
\[ \theta = 1.22 \frac{\lambda}{D} \]
where,
- \(\lambda\) is the wavelength of the radar frequency,
- \(D\) is the diameter of the radar antenna.
The wavelength \(\lambda\) can be calculated using the speed of light \(c\) and the frequency \(f\):
\[ \lambda = \frac{c}{f} \]
Given:
- Diameter of radar antenna, \(D = 2.10 \ \text{m} \)
- Frequency, \(f = 14.0 \ \text{GHz} \)
- Distance to the boats, \(d = 8.00 \ \text{km} \)
Using these formulas, you can find the minimum detectable separation \(d_{\text{min}}\).
Feel free to dive into the "Read It" or "Watch It" sections for more detailed guidance!
---
There are two buttons below the question labeled "Read It" and "Watch It," suggesting additional help materials are available either in text or video formats.
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