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(III) Another experiment you can do also uses the radius of the Earth. The Sun sets, fully disappearing over the horizon as you lie on the beach, your cyes 20 $\mathrm{cm}$ above the sand. Youimmediately jump up, your eyes now 150 $\mathrm{cm}$ above the sand, and you can again see the top of the Sun. If you count the number of seconds $(=t)$ until the Sun fully disappears again, you can estimate the radius of the Earth. But for this Problem, use the known radius of the Earth and calculate the time $t$ .

$8.8 s$

Physics 101 Mechanics

Chapter 1

Introduction, Measurement, Estimating

Physics Basics

Wan H.

July 5, 2020

University of Michigan - Ann Arbor

Hope College

McMaster University

Lectures

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

09:56

In mathematics, algebra is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form, algebra is the study of mathematical symbols and the rules for manipulating these symbols; it is a unifying thread of almost all of mathematics.

08:03

(III) You are lying on a b…

05:02

03:36

Suppose that, while lying …

03:34

(III) $(a)$ Show that if a…

01:52

The time to sunset can be …

00:35

Given radius of earth …

00:46

In the above problem, if t…

04:15

(III) Determine the mass o…

Okay. Our question states that another experiment You, Khun do also uses the radius of the earth, the sunsets fully disappearing over the horizon. As you lie on the beach, your eyes are 20 centimeters about the sand. You immediately jump up. Your eyes are now 150 centimeters above the sand and you can again see the top of the sun if you count the number of seconds and denoted as t until the sun Feli disappears again. You can estimate the radius of the earth, but for this problem, use the known radius of the earth and calculate the time T. Okay, so I have drawn here. The earth drawn the earth as a green sphere and I've been noted the radius of the earth as the line late, the redline labeled as are and the radio or the distance from our toothy new height of the eyes. After standing up, which I have denoted as H as our plus h, you can then draw a line between the eyes of the individual who is standing to the point in which the sun appears to be setting as point D. And then I also denoted point a here as this point at which the sun appears to be setting the first time. The dotted line denoted as one is the direction of the first sunset and the day not dotted line to notice. Us too, is the direction of the second sunset. The angle between these two dotted lines is data and from geometry. That's gonna be the same angle as between R and R plus H and ah, conveniently r r plus Agent d make a right triangle here, which I showed by drawing that little square in the corner where the right angle appears. So this is this is actually really useful. We can use this fact to make use of Pythagorean dirham, which states that the square of the Hy Pot news of a right triangle is equal to the sum of the square of the two sides. So the iPod news of our triangle is our play his age. So are close h squared. It's going to be equal to the sum of the two sides, so d squared plus car squared. Okay, But we can go ahead and carry out the square of our plus h. If we do that, we have our squared plus h squared is equal It I'm sorry. Plus two r h that's from carrying out the some, uh I'm sorry. That's from carrying out the square of our plus h squared. Well, another convenient fact here is the fact that h is very, very small compared to both our I'm sorry. Compared to art, so R squared is going to be significantly larger than H squared as well as two times are times H is also going to be significantly larger than H squared. So H squared here. Let's just write that fact out. So we have it visualized is much smaller, then are squared plus to our age. So those values are gonna dominate that side of the equation. So we can actually go ahead and just ignore H squared. So this gives us we're going to rewrite that degree in dear him. Now, as d squared plus r squared, send the two sides. But our high pot news now is going to be re written since we're ignoring age going to cross that out, as are squared plus two r h. All right, Well, by doing this, we see that r squared is also going to cancel out on those sides of the equation so we can go ahead and cross that off. And now we have we are able to solve for D, which we did not know before is equal to the square root two times are time's a JJ. Okay, The other thing we don't know, here is data. So we're going to make use of the Trigana metric identity of tangent. Data here is equal to the opposite of beta divided by the adjacent to theta. The adjacent being are So it's going to be opposite which is D over our well we just solved for d. We know that d is equal the square root of two r h and we know what arias. So let's go ahead and self for data. We have all the pieces in place to solve the data. So theta is going to be the inverse tangent which is denoted as tangent to the minus one of D, which is the square root of two R h would just go ahead and write it. Is that all over our If you plug in the values for R and H and you carry this out, you plug it in your calculator, you find that the fate of value is 3.66 There's a decimal place here. Times 10 to the minus two. The unit's here our degrees. So if your calculators and radiance you're not going to get that values and make sure your calculator is solving for things and units of degrees. Okay, so we ran out of room on this page. Let's go ahead and open up a another page here, and we're going to make use of the fact that it takes for one full rotation. So one rotation, this abbreviated as R. O. T for the Earth is equal to 24 hours, right? But it's also equal to 360 degrees for one full rotation, right as faras degrees goes. Okay, so we have There are conversion between the degrees and time. It takes here on Earth for one full rotation or 306 degrees to take place. So we're gonna go ahead and make use of that relationship there. But because we know fate up, we know the angle of rotation that we have relative to the full rotation or just 360 degrees so we know that ratio right. And the ratio we want to find is the ratio of the time it takes for theater to occur for that amount of rotation to occur relative to the full amount of time, which is 24 hours. Okay, but this is going to be relative relatively small, maybe on the order of a few seconds relative to that 24 hours. Let's go. Heading for hours in the seconds toe work with more relevant nicer units. So 24 hours home R h r not just are multiplied by the amount of seconds and one hour, which is 36 100 seconds per hour will give us time in 24 hours, which comes out to be 86,000 400 seconds. Okay, so now we have everything we need to solve for time T. So if we solve for T, we see that tea is equal to data over 360 degrees multiplied by our 86,400 seconds. Okay. And that comes out to be 8.8 seconds. So that's how much time passes between the first in the second since it all right in that box it in because that is our solution.

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