Relative Motion Review— Sine and Cosine Law Solution

Summary

The video explains how to determine the heading and airspeed of a plane navigating in windy conditions. The scenario involves a pilot flying to a city 600 km away in two hours, with wind blowing at 70 km/h at 10° west of south. Through vector analysis, diagrams, and calculations using trigonometric laws (cosine and sine laws), the airspeed of 278 km/h and a heading of 1.81° south of west are determined. The process includes understanding resultant vectors and correcting for wind direction to achieve the desired flight path.

Takeaways

• 😀 The pilot's goal is to reach a city 600 km away in 2 hours, heading 15 degrees south of west.
• 🌬️ There is a 70 km/h wind blowing at 10 degrees west of south, which affects the plane's trajectory.
• 🛫 The plane's resultant velocity, considering the wind, is 300 km/h, calculated by dividing the distance by the time.
• 🔍 The script uses vector addition to determine the plane's heading and airspeed, considering the wind's effect.
• 📏 The cosine law is applied to find the plane's airspeed by using the known wind speed and resultant velocity.
• 🧮 The calculated airspeed of the plane is approximately 278 km/h, accounting for the wind's influence.
• 📐 The sine law is used to determine the angle between the wind vector and the plane's heading.
• 🔄 The plane's heading is found by subtracting the angle calculated using the sine law from the desired direction.
• 🏁 The final heading of the plane is approximately 1.81 degrees south of west, adjusted for the wind.
• 📘 The script provides a detailed step-by-step approach to solving the problem using trigonometric laws and vector analysis.

Q & A

What is the distance the pilot needs to cover to reach the city?

The pilot needs to cover a distance of 600 kilometers to reach the city.

In which direction does the pilot need to fly to reach the city?

The pilot needs to fly in a direction 15 degrees south of west to reach the city.

What is the speed and direction of the wind?

The wind is blowing at a speed of 70 kilometers per hour at a direction of 10 degrees west of south.

How long does the pilot have to reach the city?

The pilot has two hours to reach the city.

What is the resultant speed of the plane required to reach the city in the given time?

The resultant speed of the plane is 300 kilometers per hour, calculated by dividing the distance by the time.

What is the angle between the wind direction and the desired direction?

The angle between the wind direction (10 degrees west of south) and the desired direction (15 degrees south of west) is 65 degrees.

What is the formula used to calculate the unknown side of the triangle in this scenario?

The cosine law is used to calculate the unknown side of the triangle, which is represented as 'b^2 = a^2 + c^2 - 2ac cos(B)'.

What is the calculated airspeed of the plane with respect to the air?

The calculated airspeed of the plane is 278 kilometers per hour.

How is the heading of the plane determined?

The heading of the plane is determined by subtracting the angle 'a' from the desired direction angle of 15 degrees south of west.

What is the final heading of the plane after accounting for the wind?

The final heading of the plane after accounting for the wind is 1.81 degrees south of west.

What trigonometric function is used to find the angle 'a' in the triangle?

The sine function is used to find the angle 'a' in the triangle, using the formula 'sin(a) = opposite side / hypotenuse'.

Outlines

00:00

🛫 Introduction to the Vector Problem of Airplane Heading with Wind

In this paragraph, the problem of determining the heading and airspeed of a plane flying towards a city 600 km away in 2 hours is introduced. The wind is blowing at 70 km/h, 10 degrees west of south, and the goal is to find the plane's required heading and airspeed to compensate for the wind. A diagram is referenced to illustrate the vector relationship between the airspeed, wind, and resultant velocity needed to reach the city in the desired direction, which is 15 degrees south of west.

05:00

✈️ Analyzing the Angles in the Problem Using Parallel Lines

This section focuses on determining key angles in the vector diagram using parallel lines and alternate interior angles. First, the angle between the resultant velocity and the east-west line is identified as 80 degrees. Then, the smaller angle of 15 degrees south of west is noted. By subtracting these two angles, a 65-degree angle in the triangle is calculated. This angle will be used later to solve for the plane's airspeed using the cosine law.

10:03

📐 Solving for Airspeed Using the Cosine Law

In this paragraph, the cosine law is applied to find the magnitude of the airspeed vector. The sides and angles of the triangle are labeled, and the relevant equation for cosine law is set up. Using the given wind speed (70 km/h), the resultant velocity (300 km/h), and the calculated angle (65 degrees), the airspeed of the plane is found to be 278 km/h. This value represents the speed the plane must fly to compensate for the wind.

🎯 Determining the Plane’s Heading Using the Sine Law

This paragraph explains how to calculate the plane's heading using the sine law. With known values of angle B (65 degrees) and side B (278 km/h), along with side A (70 km/h), angle A is found to be approximately 13.19 degrees using the inverse sine function. Finally, the heading is calculated by subtracting this angle from 15 degrees, resulting in a heading of 1.81 degrees south of west. The paragraph concludes with the final solution: the airspeed of 278 km/h and a heading of 1.81 degrees south of west.

Keywords

💡Relative motion

Relative motion refers to the concept that the movement of an object is always measured in relation to another object or reference frame. In the video, this concept is applied to determine the direction and speed of a plane considering the wind's influence. The goal is to find the actual movement (resultant vector) based on the relative motion of the plane and the wind.

💡Resultant vector

A resultant vector represents the combined effect of two or more individual vectors, such as direction and speed. In the video, the resultant vector shows the actual path and velocity the pilot wants to achieve, which combines the plane's heading and airspeed with the wind's direction and speed.

💡Heading

Heading refers to the direction in which a plane or vehicle is pointed or traveling. In the video, the heading is what the pilot must adjust to counteract the wind and reach the intended destination at 15 degrees south of west.

💡Wind speed

Wind speed is the velocity at which the wind is blowing in a specific direction. Here, it’s crucial because the wind, blowing at 70 km/h and 10 degrees west of south, affects the plane’s actual motion, and the pilot needs to account for this when determining the correct heading.

💡Cosine law

The cosine law is a trigonometric formula used to calculate the unknown side or angle of a triangle when two sides and the included angle are known. In the video, this law is used to find the airspeed of the plane by applying it to a triangle formed by the wind speed, resultant velocity, and airspeed.

💡Sine law

The sine law is a mathematical formula used in trigonometry to relate the angles and sides of a triangle. In the video, the sine law helps determine the plane's heading by comparing the ratio of the known angle and opposite side to the unknown angle and its corresponding side.

💡Airspeed

Airspeed is the speed at which an aircraft moves relative to the air around it. The airspeed is a critical factor in the video because it needs to be adjusted for the wind to achieve the desired resultant velocity of 300 km/h toward the target city.

💡Degrees south of west

This term refers to the direction an object is moving relative to the cardinal directions. For example, '15 degrees south of west' means that the plane's intended path is slightly southward from a straight westward direction. This heading is the pilot’s target direction in the video.

💡Vector addition

Vector addition involves combining multiple vectors to find a single resultant vector. In the video, the pilot's airspeed and the wind speed are added as vectors to find the actual movement of the plane, showing how the plane needs to adjust its heading to reach its destination.

💡Parallel lines and angles

Parallel lines and their associated angles are used in geometry to simplify complex angle relationships. In the video, parallel east-west lines are used to determine the angles formed by the wind and the desired resultant path, helping the pilot calculate the correct heading to reach the destination.

Highlights

The pilot aims to reach a city 600 kilometers away in two hours, facing a wind blowing at 70 km/h at 10 degrees west of south.

The desired direction to the city is 15 degrees south of west.

The resultant velocity is calculated as 300 km/h, derived from the distance and time required to reach the destination.

The wind vector is described as 80 degrees south of west for simplification in calculations.

Parallel lines and alternate interior angles are used to determine the angles in the vector diagram.

The angle between the wind vector and the desired direction is calculated to be 65 degrees.

The cosine law is applied to find the airspeed of the plane without wind.

The airspeed is calculated to be 278 km/h using the cosine law.

The heading of the plane is determined by subtracting the angle 'a' from the desired direction angle.

The sine law is used to find the angle 'a' based on the known sides and angles in the triangle.

The angle 'a' is found to be approximately 13.19 degrees using the inverse sine function.

The final heading of the plane is calculated to be 1.81 degrees south of west.

The airspeed vector is determined to be 278 km/h at a heading of 1.81 degrees south of west.

The solution involves correcting for the wind's effect on the plane's trajectory.

The problem-solving process is visualized through a diagram that helps in understanding the vector addition.

Transcripts

this is question two from the vector

problems that have sine and cosine lots

of solutions from the review for

relative motion

so a pilot wishes to reach a city 600

kilometers away in the direction 15

degrees south of west

in two hours

if there is a wind that is below of 70

kilometers per hour

blowing at 10 degrees west of south

what must be the heading and air speed

of the plane

so i've drawn a diagram that goes from a

to b

right and it's in a direction 15 degrees

south of west which i've

labeled here

so we want to find the heading and the

air speed of the plane

so since we know that we want to get

from a to b we want to go in a direction

15 degrees south of west

that must mean that this

is the resultant this is what

someone on the ground would see this is

where they would actually end up going

so that must mean that the pilot has to

correct for the wind speed so we want to

find what the direction and the speed

that the plane would go in

if there was no wind

that is the heading and air speed

so that must mean that our heading and

airspeed our airspeed vector

has to add to the wind to find our

resultant in blue here

so the wind is blowing at 10 degrees

west of south so that's going to look

like this

10 degrees west of south and then our

air speed so our original speed

um is going to be

from here to here

so you can see that this is our

air speed

and it adds to the wind speed

and gives us our resultant

okay so i've relabeled the diagram to

make it a little bit neater

this is our wind speed this

70 kilometers per hour here

i've just labeled the

80 degrees south of west instead of 10

degrees west of south just to make it a

little bit neater

okay so let's take a take a look at what

we

know and what we need to know so they

tell us that

the wind speed is 70 kilometers per hour

but we actually also know the resultant

speed and that is because we know

the distance that we have to travel

and how fast we need to get there

so that is our resultant because that is

how fast that we actually want to go

so

our velocity is going to be distance

divided by time

which is going to be 600 kilometers

divided by 2 hours

that is going to give us 300 kilometers

per hour this is our resultant speed

because this is how fast we want to go

okay so let's label that 300 kilometers

per hour

here

okay so now we have to look for some

angles and in order to find angles we

need parallel lines

so

what i see first is this set of parallel

lines so if you take a look at this

east-west line and this east-west line

they're parallel and this

our wind vector is kind of like a

diagonal

and we know

that our alternate interior angles which

would be this

80 degrees here

and this

big angle so from the east-west line all

the way up to that

vector

they are the same so we know that this

big vector is 80 degrees

now let's look at another pair

of parallel lines

let's look at

this east-west line

and

this

oops

this east-west line

okay

so these two are parallel now if we take

a look at

this diagonal

right

we know this angle here the alternate

interior is going to be this angle here

so they're going to be the same so we

know that this angle is 15 degrees

so

if we want

um

if we want

this angle

then

we can

take 80 degrees which is this

this bigger one minus

15 degrees which is this smaller one

and 80 minus 15 that would give us 65

degrees and that is our angle

oops

our angle that we want

in the triangle so this is

65 degrees

in the triangle here

okay so now that we've found this angle

65 degrees

we can find our

air speed which is going to be this

purple line here

okay so i will i've labeled the

angles

so i called this one angle a i'll call

this one angle b and i'll call this one

angle c

okay so

i'll label the sides as well so i'll

call this one

side c since it's opposite to angle c

and i'll call this one side a since it's

opposite angle a

and our last one which is what we're

looking for is side b and we don't know

that yet

okay so if we take a look at what we

know we have this side length

we have this side length

and we have an angle that is in between

them

and since this angle happens to be the

angle that is opposite

to side b which is what we're looking

for we can use the cosine law

so let's set up the cosine law

and

the letters that are used in the

regular format of the cosine law that

doesn't really matter

as long as you have two sides and an

angle in between so

now my cosine law is going to look like

b squared equals a squared plus c

squared minus 2

a c

cosine

b

where this

is the angle

opposite

to the side that we're looking for so

the letter names don't really matter

as long as you have two sides and an

angle between them

okay so let's solve for b

b is going to equal the square root

of

all of this so a is 70

70 squared plus 300 squared which is

going to be our c squared

minus 2 times

a sorry 2 times a

times c

times cosine b which is the angle across

from the side that we don't know so

across from the angle that don't know 65

degrees

and if you plug this entire expression

into your calculator

that should give you

278

kilometers per hour if we're rounding to

three sig figs

okay

so now we know that b is 278

kilometers per hour

and now

we just need to look for the heading so

to look for the heading

we need an angle so you can look for

this angle

or you can look for

this angle

i am going to stick to

this top angle here that i've labeled in

blue

okay so to do that we can find angle a

and then we can subtract angle a from

15. so we do 15 minus angle a that will

give us our heading

so let's do that to find angle a

we can use the sign law

since we have the side a that's across

from it we have another angle

and we have the side that is across from

that angle so let's set it up

we will have

sine a

over a which is the side that's opposite

to it

equals sine b

over b to the side opposite to that so

sine a

we don't know what angle a is but we do

know the side that is opposite to it

right here

a equals 70 kilometers per hour so sine

a divided by 70 equals sine b

which is the angle that we have here 65

degrees

sine 65 divided by

side b which is across from it and we

just solve for side b side b right

that's 278.

all right so let's isolate sine a

so sine a equals 70 times sine 65

divided by 278 so i've just multiplied

the 70 over

and that will give us

that will give us

0.228

i'm not going to round here since we

still need to find angle a and to find

angle a we need to take the inverse sine

of this decimal so

inverse sine of

0.228

and that will give us

13.19 something degrees now be careful

angle a is not our final answer but

instead we need the heading which again

is going to be 15 degrees minus a

so our heading will be

our

heading

is equal to 15 degrees

minus a which is 13.19

something degrees

which will give us

1.81 degrees if we just round to three

sig figs

so our final velocity velocity of the

plane with respect to the air so our air

speed

is going to be

our magnitude which are which we

calculated here

278 kilometers per hour

at

1.81 degrees now what direction is this

well if

this

is west

we are going

to the south of west

so we are going to go

1.81 degrees

south

of

west