Finding the shortest distance between two places on earth.

Summary

In this educational video, the host teaches viewers how to calculate the shortest distance between two points on Earth's surface using great circle navigation. The example involves towns A and B, located at 60 degrees North and 42 degrees West, and 60 degrees North and 29 degrees East respectively. The process includes determining the central angle subtended by the arc connecting the two points, using the Earth's radius of 6370 km. The video concludes with the formula and calculation resulting in a distance of approximately 3754 kilometers between the towns.

Takeaways

• 🌐 The video is about determining the shortest distance between two points on Earth's surface using longitudes and latitudes.
• 📍 The towns A and B are located at 60 degrees north and 42 degrees west, and 60 degrees north and 29 degrees east, respectively.
• 🌍 The shortest distance between two points on Earth is along a great circle, which is a circle with a radius equal to the Earth's radius.
• 📈 The Earth's radius is taken as 6370 kilometers for the calculation.
• 🔍 The towns A and B are located on the same latitude but have different longitudes, creating a longitudinal difference.
• 📐 The angle subtended by the arc between A and B at the Earth's center, denoted as X, is crucial for calculating the distance.
• 🧭 The formula to find the arc length is X/360 * 2πr, where r is the Earth's radius.
• 📈 The value of X is determined using the formula: sin(x/2) = cos(Alpha) * sin(Theta/2), where Alpha is the latitude and Theta is the longitudinal difference.
• 🔢 The longitudinal difference is calculated as 71 degrees by adding 42 degrees west and 29 degrees east.
• 📊 After calculating, the angle X is found to be approximately 33.758 degrees.
• 📝 The final calculation of the shortest distance between the two towns is approximately 3754 kilometers.

Q & A

What is the main topic of the video?

The main topic of the video is to determine the shortest distance between two points on the surface of the Earth using longitudes and latitudes.

What are the two towns whose shortest distance is being calculated in the video?

The two towns are Town A located at 60 degrees North 42 degrees West and Town B at 60 degrees North 29 degrees East.

What is the radius of the Earth used in the calculation?

The radius of the Earth used in the calculation is 6370 kilometers.

Why is the distance along a great circle considered the shortest between two points on Earth's surface?

The distance along a great circle is considered the shortest because it represents the shortest path on the surface of a sphere, which is the Earth's shape.

How is the angle subtended at the center of the Earth (X) calculated in the script?

The angle X is calculated using the formula: sine of X over 2 equals cosine of Alpha times sine of Theta over 2, where Alpha is the latitude and Theta is the longitudinal difference.

What is the value of the latitude (Alpha) used in the calculation?

The value of the latitude (Alpha) used in the calculation is 60 degrees North.

How is the longitudinal difference (Theta) determined in the script?

The longitudinal difference (Theta) is determined by adding the absolute values of the longitudes of the two towns: 42 degrees West and 29 degrees East, resulting in 71 degrees.

What is the formula used to calculate the length of the arc AB, which represents the shortest distance between the two towns?

The formula used to calculate the length of the arc AB is X over 360 times 2 pi r, where X is the angle subtended at the center of the Earth, pi is approximately 22/7, and r is the radius of the Earth.

What is the calculated shortest distance between Town A and Town B in kilometers?

The calculated shortest distance between Town A and Town B is approximately 3754 kilometers.

What is the significance of the equator and the prime meridian in locating the towns on Earth's surface?

The equator and the prime meridian are significant as they serve as reference points for latitude and longitude, respectively, allowing for precise location of any point on Earth's surface.

How does the video script illustrate the concept of a great circle?

The video script illustrates the concept of a great circle by showing that the shortest distance between two points on a sphere is along the arc of a circle with the same radius as the sphere, which in this case is the Earth.

Outlines

00:00

🌐 Determining Shortest Distance on Earth's Surface

This paragraph introduces the video's objective to teach viewers how to calculate the shortest distance between two points on Earth's surface, specifically between Town A at 60°N 42°W and Town B at 60°N 29°E. It emphasizes the use of great circles, which are the shortest paths on a sphere, and sets the stage for the mathematical approach that will be taken to find the distance, considering the Earth's radius as 6370 kilometers.

05:00

📐 Calculating the Great Circle Distance

The second paragraph delves into the process of calculating the shortest distance between the two towns using the great circle method. It explains the need to determine the central angle (X) subtended by the arc connecting the two points at the Earth's center. The formula to find the length of the arc (AB) is introduced, which involves the angle X, the Earth's circumference, and its radius. The paragraph then guides through the calculation of the central angle using the latitude and the longitudinal difference between the two towns, resulting in an angle of approximately 33.758°. Finally, it applies the formula to compute the shortest distance, which is approximately 3754 kilometers.

Keywords

💡Longitudes and Latitudes

Longitudes and latitudes are the angular measurements used to determine a location on the Earth's surface. Longitudes run from the Prime Meridian and measure east or west, while latitudes measure north or south from the equator. In the video, these terms are fundamental to understanding how to calculate the shortest distance between two points, Town A and Town B, using their respective coordinates.

💡Shortest Distance

The shortest distance between two points on the Earth's surface, especially when not in a straight line, is along a great circle. The video focuses on calculating this distance between two towns using their geographical coordinates. The concept is crucial for navigation and understanding the Earth's spherical geometry.

💡Great Circle

A great circle is the largest circle that can be drawn on a sphere, with its center at the Earth's core. The video explains that the shortest path between two points on the globe is along this circle. The great circle is significant in the context of the video as it is the basis for the formula used to calculate the distance between Town A and Town B.

💡Prime Meridian

The Prime Meridian is the longitudinal line at 0 degrees, which serves as the reference point for measuring longitudes east and west. In the video, it is used as a starting point to determine the longitude of Town A and Town B, which is essential for calculating their positions and the distance between them.

💡Equator

The equator is the imaginary line around the Earth at 0 degrees latitude, dividing the Earth into the Northern and Southern Hemispheres. In the script, it is mentioned as a reference for measuring latitude, which is important for locating Town A and Town B on the Earth's surface.

💡Arc

An arc is a segment of a circle. In the context of the video, the arc refers to the portion of the great circle that connects Town A and Town B. The length of this arc is the shortest distance between the two towns, which is the main focus of the video's calculation.

💡Radius of the Earth

The radius of the Earth is the distance from the Earth's center to its surface, approximately 6370 kilometers. The video uses this measurement as a constant in the formula to calculate the shortest distance between the two towns, emphasizing its importance in spherical geometry calculations.

💡Angle Subtended

In the video, the angle subtended refers to the angle at the center of the Earth between the lines drawn from the center to Town A and Town B. This angle is crucial for determining the length of the arc, which represents the shortest distance between the two points.

💡Sine and Cosine

Sine and cosine are trigonometric functions used in the calculation of angles in a right-angled triangle. The video uses these functions in the formula to find the angle subtended at the Earth's center, which is necessary to calculate the shortest distance between Town A and Town B.

💡Trigonometric Formula

The trigonometric formula used in the video combines sine and cosine functions to relate the angle subtended at the Earth's center with the latitude and the longitudinal difference between the two towns. It is essential for finding the central angle that, when used in the arc length formula, gives the shortest distance.

💡Spherical Geometry

Spherical geometry is the study of geometric shapes on the surface of a sphere. The video is an application of spherical geometry, as it involves calculating distances on the Earth's surface using the properties of great circles and angles.

Highlights

The video teaches how to determine the shortest distance between two points on Earth's surface using longitudes and latitudes.

The shortest distance between two points on Earth is along a great circle.

The radius of the Earth is taken as 6370 kilometers for this calculation.

Town A is located at 60 degrees north latitude and 42 degrees west longitude.

Town B is at the same latitude but 29 degrees east longitude.

The shortest distance is found by considering the great circle passing through both towns and the Earth's center.

The angle subtended by the arc at the Earth's center, denoted as X, needs to be calculated.

The formula to find the arc length is X/360 * 2πr, where r is the Earth's radius.

Latitude and longitude differences are used to calculate the value of X.

The latitude difference (Alpha) is 60 degrees north for both towns.

The longitude difference (Theta) is calculated as 42 degrees west + 29 degrees east = 71 degrees.

The formula to find X is sine(x/2) = cosine(Alpha) * sine(Theta/2).

X is calculated using the inverse sine function and the values of Alpha and Theta.

The calculated angle X is 33.758 degrees.

The shortest distance formula is applied using the calculated angle X to find the arc length.

The final calculated shortest distance between Town A and B is approximately 3754 kilometers.

The video concludes by summarizing the method and result for finding the shortest distance between two points on Earth.

Transcripts

[Music]

foreign

[Music]

welcome back to our Channel

in this video we're going to learn how

to determine the shortest distance

between two points on the surface of the

Earth

now that is under the topic longitudes

and latitudes

so the question we have here is

determine the shortest distance between

towns a 60 degrees north 42 degrees west

and P 60 degrees North 29 degrees east

in kilometers take the radius of the

Earth as

6370 kilometers

so the first thing we need to note that

the shortest distance between two points

on the surface of the Earth is the

distance along a great circle

let us locate tone A and B

on the surface of the Earth

so you have this sketch

we have two reference points that is the

prime meridian

usually set at zero degrees

and you have the equator also set at

zero degrees

now tone a lies

60 degrees north 42 degrees west

so we have it lying on latitude 60

degrees

north of the equator

and

42 degrees west

of the prime meridian so how

42 degrees west

now where the longitude and the latitude

meet that is the location of term a

now Photon B Town B lies on the same

latitude 60 degrees north but 29 degrees

east of the Prime Meridian that is this

longitude here

29 degrees

to the east of the Prime Meridian

so where are the two meet that is

longitude 29 degrees east and latitude

60 degrees north that is the location of

town B

now just as I had mentioned earlier on

the shortest distance between these two

towns will be the distance along a great

circle

and we are going to consider

points a here to be the center of the

Earth

so that if we join

a to the center of the earth and the B

to the Center of the Earth

then we can come up with acceptor

by joining a to B in this manner

from an arc there so we have this sector

here now this Arc joining A and B which

is part of this sector is the shortest

distance now we make it the shortest

distance because it is a part of a great

circle so we have this great circle I'll

have dotted

now we call it a great circle because

its radius is the radius of the Earth so

this radius here is 6370 kilometers

now in order to determine

the shortest distance that is the length

of this Arc a b

we need to know the angle it subtends at

the center of the earth let's call it X

well before we get that X we need to

take note that

the distance a b

considering this sector if we have the

center of the earth as

or so consider the sector o a b

in order to get the length a b that is

the act length

then we'll use the formula

X over 360.

times 2 pi r

and therefore our task here is to

determine the value of x and x will be

determined from this formula so we say

sine of x over 2

is equal to cosine of alpha

sine of theta over 2.

so

we'll take note that Alpha here

represents

the latitude

and Theta represents longitudinal

difference

so

first thing to determine is the value of

the latitude and the longitudinal

difference so

the latitude from the question we've

seen is 60 degrees north

and the longitudinal difference will

determine as follows so we have

the prime meridian and then to the east

of the Prime Meridian we have this

longitude 29 degrees that is east of the

Prime Meridian note that the Prime

Meridian is usually set at zero degrees

so to the east

of the Prime Meridian we have longitude

29 degrees east and then

to the West we have longitude 42 degrees

west so the angle difference between

longitude 42 degrees west and longitude

29 degrees east is the sum of 42 and 29

so that is from 42 degrees west to the

prime meridian here we have 42 degrees

and from the prime meridian to longitude

29 degrees east we have 29 degrees so

the total angle between the two

longitudes is just the sum so we're

going to have Theta as 42

plus 29

so that is 71 degrees

with this we can now determine the value

of x from this formula so we'll do our

substitutions so we'll say

sine of x over 2

is equal to cosine of Alpha and Alpha is

60. times sine of theta over 2

and Theta is 71 and therefore we divide

by 2 and that is 35.5

so from here when we work out the right

hand side we are going to get 0.29

035 from my calculator I'm able to get

that

and therefore

this means that X over 2 is the sine

inverse of

0.29035

and that is

so we have 16.879

and then

because we're looking for X remember

this is X over 2 and therefore in order

to get X multiply both sides by 2 and

therefore X is 33.75

8.

so this is the angle subtended

by this arc Arc a b

which is the shortest distance between

Town A and B

so we're going to use it in this formula

here

so let's write it

a b

is equal to X over 360 and X is the

3.758 all over 360. times

2 times pi and Pi is 22 over 7 times r r

is the radius of the Earth and that is

6370.

so

from my calculator when I work out these

I'm able to get three thousand seven

hundred and fifty four

points

four

this is approximately

three thousand seven hundred and fifty

five

kilometers

and finally we have the shortest

distance between the two terms just as

had been desired in the question

thank you for watching see you in the

next one

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