A radar station is located at a height of 24.0 m above the shoreline. When the radar is aimed a

A radar station is located at a height of 24.0 m above the...

Key Concepts

Speed of Light

The speed of light is the constant rate at which electromagnetic waves travel through a vacuum, approximately 3.00 × 10^8 m/s. This constant is fundamental in calculating distances based on the time it takes for a light (or radar) signal to travel from a source to an object and back.

Time-of-Flight Measurement

Time-of-flight measurement is a technique where the time interval between emitting a signal and receiving its reflection is recorded. This measured time, when combined with the known speed of the signal, allows for the determination of the distance to the reflecting object by multiplying the travel time by the speed of light.

Round-Trip Calculations

In many practical applications, including radar, the measured time represents the total time taken for a signal to travel to the target and back. To find the one-way distance, it is necessary to divide the total travel distance by two. This concept is essential when converting a round-trip time measurement into an actual distance from the source to the target.

Right Triangle Geometry

Right triangle geometry is used in scenarios where the distance between two points involves both horizontal and vertical components. By forming a right triangle, the Pythagorean theorem can be applied to relate the direct (hypotenuse) distance with its horizontal and vertical components, allowing for the extraction of unknown lengths such as depth.

Transcript

00:01 Radar or radio waves is an example of an extremely long wavelength of an electromagnetic wave that travels at the speed of light.

00:14 So just like light will refract when going into water, so will radar.

00:22 And here we have a situation where we will be using snell's law, but with radar.

00:28 But we will assume that the index of refraction of the materials through which the radar propagates are the same as visible light to a first order that is probably a good approximation.

00:45 But snell's law relates the index of refraction on one side of the boundary here between air and water times the sign of the angle to the normal, that's important, is equal.

01:00 To the product of the index times the sign of the angle to the normal on the other side.

01:09 Here we have relatively complex geometry, and we're also going to need the fact that the index of refraction measures how much something slows down upon entering different material in air or vacuum.

01:27 The speed of light c is 3 times 10 to the 8th, roughly, meters per second.

01:37 Actually, i'll be using a little bit more significant figures just to be a little bit more accurate.

01:45 So i'm going to use 2 .998, but it is probably okay to use 3 in many instances.

02:03 And here we see that n measures how much the material slows down the light.

02:13 Why we will need that is because we are told the return trip time for a radar signal to go all the way from a radar station out to a point on the ocean surface and then down under, striking a whale, scattering off the whale, and coming all the way back.

02:34 So here i'll write down that the total time is equal to 2 .1 microseconds for a one -way trip, delta t1 way, i'll just call that delta t1, is half of that.

02:55 And of course, they give us the round trip because that is what the station measures.

03:04 But we are going to assume that it took the same time to return as it did to get back, go all the way out there and hit the whale.

03:16 All righty.

03:16 So a lot of things to put together, but we have enough geometry so that we can get the sign of theta 1.

03:26 We have some distances here.

03:30 Tangent of theta 1, for example, we could find through 20, let's see, 150 over 24, and we thus get theta 1 as 80 .9 degrees.

04:05 80 .91, to be exact, carry things to four sick figs until i'm done calculating because there are number of calculations.

04:16 So now we can implement snells law, and we can find the angle on the other side of the water.

04:27 Okay, so we have 1 .00.

04:32 We'll use that for air, times sign.

04:37 Yeah, pretend we have good old air there.

04:40 Sign of 80 .91 degrees divided by n2.

04:53 We're going to use the n for water is equal to sine of theta 2.

05:03 Again, we'll keep some more significant figures than we probably need, but we do know the index of refraction quite accurately for both air and water.

05:25 And let's see, we will take the inverse sign of the numbers, and we get that theta 2 is 47 .8 degrees.

05:43 Okay, now what we're really after is the depth of this whale, which is supposedly sitting at the bottom of the ocean.

05:52 So they're using the whale as kind of a scattering target, if you will...

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