What is the maximum distance a radar can detect objects on the Earth's surface?

[utkarshakash]

Homework Statement

A radar has power 1kW and is operating at a frequency of 10GHz. It is located on a mountain top of height 500m. The maximum distance upto which it can detect object located on the surface of the earth(Radius of Earth = 6.4*10^6 m) is

Homework Equations

The Attempt at a Solution

I'm not sure what to do here because I'm facing this kind of problem for the first time. Can anyone suggest me some equations to start with?

 

[Rellek]

D = sqrt(2*height*radius) I think

 

[utkarshakash]

D = sqrt(2*height*radius) I think

This gives me the correct answer but how did you think of this? Can you give some references regarding this equation?

 

[BvU]

You google Distance to Horizon

 

[HalfLostAndSearching]

I can provide some insights on how to approach this problem. First, we need to understand the basics of radar technology. Radar stands for "radio detection and ranging" and it works by sending out radio waves and then receiving the reflected waves from objects in its path. The time it takes for the waves to return can be used to calculate the distance to the object.

In this case, we are given the power of the radar (1kW), its operating frequency (10GHz), and its location (500m above the ground). We can use these values to calculate the maximum distance the radar can detect objects on the surface of the Earth.

To start, we can use the formula for the speed of light (c) in a vacuum, which is approximately 3*10^8 m/s. This speed is constant and is the speed at which radio waves travel.

Next, we can use the formula for the wavelength (λ) of a wave, which is equal to the speed of light divided by the frequency. In this case, the wavelength would be 3*10^8 m/s divided by 10*10^9 Hz, which is equal to 0.03 meters.

Now, we can use the concept of radar's "line of sight" to calculate the maximum distance it can detect objects. Since the radar is located 500m above the ground, we can draw a right triangle with the radar at the top, the object on the surface of the Earth at the bottom, and the hypotenuse being the maximum distance the radar can detect.

Using the Pythagorean theorem, we can calculate the length of the hypotenuse, which is equal to the square root of (500m)^2 + (maximum distance)^2. We can substitute the wavelength (0.03 meters) for the maximum distance and solve for it.

The final equation would be √(500m)^2 + (0.03m)^2 = maximum distance.

Solving this equation, we get a maximum distance of approximately 500.001 meters. This means that the radar can detect objects on the surface of the Earth up to a distance of 500 meters.

I hope this explanation helps you understand how to approach this problem. Remember, as a scientist, it is important to understand the basic concepts and equations related to a problem before attempting to solve it. Good luck!