Trigonometry and Vectors Motion and Forces in Two Dimensions SP1b. Compare and constract scalar and vector quantities

SP1b. Compare and constract scalar and vector quantities.

Trigonometry and Vectors

Motion and Forces in Two Dimensions


Standard

SP1. Students will analyze the relationships between force, mass, gravity, and the motion of objects.

b. Compare and contrast scalar and vector quantities.


Right Triangles

• The longest side is the hypotenuse. It is opposite the 90º angle.

• The other two sides are named depending on where they are in relation to the angle you are looking at.


The Pythagorean Theorem

• For any right triangle, the sum of the areas of the two small squares is equal to the area of the larger.


Animation


Formula

• The formula we use is:a2 + b2 = c2

• C = the length of the hypotenuse• The other two sides are a and b.


We Also Need to Know the Angles• Acute angle A is drawn in

standard position as shown.

Right-Triangle-Based Definitions of Trigonometric FunctionsFor any acute angle A in standard position,

adjacent side

opposite sidetan

hypotenuse

adjacent sidecos

hypotenuse

opposite sidesin

x

yA

r

xA

r

yA

opposite side

adjacent sidecot

adjacent side

hypotenusesec

opposite side

hypotenusecsc

y

xA

x

rA

y

rA


Easy Way To Remember

Example Find the values of sin A, cos A, and tan A

in the right triangle.

Solution– length of side opposite angle A is 7– length of side adjacent angle A is 24– length of hypotenuse is 25

247

tan,2524

cos,257

sin AAA


VECTORS AND VECTOR RESOLUTION


Scalar


Vector


Vectors


Vector Addition

• VECTOR ADDITION – If 2 similar vectors point in the SAME direction, add them.

• Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started.


Vector Subtraction

• VECTOR SUBTRACTION - If 2 vectors are going in opposite directions, you SUBTRACT.

• Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started.


More Examples


Angle Direction


Vectors Are Typically Drawn to Scale


So How Do We Add These?


PYTHAGOREAN THEOREM


Example


Example

• Eric leaves the base camp and hikes 11 km, north and then hikes 11 km east. Determine Eric's resulting displacement.


PARALLELOGRAM METHOD


You Might Find this Useful


USING TRIG FUNCTIONSDetermining Direction

Vectors can be broken down into components

A fancy way of saying how far it goes on the x and on the y.

We calculate the sides using trig.


"A" is used to represent the vector.

cos

cos

cos

x

x

A

HA

AA A


Now the Other Side

sin

sin

sin

y

y

O

HA

AA A


Example

• What if you have a vector that is 45 m @ 25o?

cos

45cos 25

40.8

x

x

x

A A

A

A m

sin

45sin 25

19.0

y

y

y

A A

A

A m



So How Do We Find Value of Direction?

• The direction is given as an angle from the +x axis

• Positive is counterclockwise

1

tan

tan

tan

y

x

y

x

O

AA

A

A

A


GRAPHICAL METHOD (TIP TO TAIL)


Graphical Method


The Order Doesn’t Matter

• Same three vectors, different order


Animation


Resultants





• (100 km/hr)2 + (25 km/hr)2 = R2

• 10000 km2/hr2 + 625 km2/hr2 = R2

• 10625 km2/hr2 = R2

• SQRT(10 625 km2/hr2) = R• 103.1 km/hr = R


• tan (theta) = (opposite/adjacent)• tan (theta) = (25/100)• theta = invtan (25/100)• theta = 14.0 degrees


Animation


Steps in vector addition

1. Sketch the vector– Head to tail method

– Parallelogram method

2. Break vectors into their components

3. Add the components to calculate the components of the resultant vector

4. Calculate the magnitude of R

5. Calculate the direction of R

Add 180o to q if Rx is negative


Example• A boat moves with a velocity of 15 m/s, N in a

river which flows with a velocity of 8.0 m/s, west. Calculate the boat's resultant velocity with respect to due north.


Example• A plane moves with a velocity of 63.5 m/s at

32 degrees South of East. Calculate the plane's horizontal and vertical velocity components.

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Trigonometry and Vectors Motion and Forces in Two Dimensions SP1b. Compare and constract scalar and vector quantities