Question

A circular radar antenna on a Coast Guard ship has a diameter of 2.10 m and radiates at a frequency of 11.0 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

Answer 1

Given Information:  

Frequency = f = 11 Ghz = 11x10⁹  Hz

Distance between boats = L = 8 km = 8000 m

Diameter of radar antenna = D = 2.10 m

Required Information:  

Distance between boats = d = ?

Answer:  

Distance between boats = d = 126.65 m

Explanation:  

The minimum distance between the two boats that will result in the detection by the radar antenna is given by

d = Ltan(θmin)

Where θmin is given by

θmin = sin⁻¹(1.22*λ/D)

Where D is the diameter of the radar antenna and λ is the wavelength given by

λ = c/f

Where c is the speed of light

λ = 3x10⁸/11x10⁹

λ = 0.02727 m

θmin = sin⁻¹(1.22*(0.02727)/2.10)

θmin = 0.907°

Therefore, the minimum distance between the two boats so that the radar antenna may detect is

d = 8000*tan(0.907)

d = 126.65 m