Introduction to Vectors2D KINEMATICS I

Vectors & Scalars

A scalar quantity has magnitude but no direction

Examples:

speed

volume

temperature

mass

A vector is a quantity that has both magnitude and direction

Examples:

displacement

velocity

acceleration

force

Vectors in One Dimension

With Motion in 1D, our vectors could point in only two possible directions:

positive

Negative

Direction was indicated by the sign (+/-)

Examples: +10 m/s meant “to the right” or “up”

9.81 m/s2 meant “to the left” or “down”

Vectors in Two Dimensions

In 2D motion, objects can move

up-down and left-right

north-south and east-west

Direction must now be specified as an angle

Vectors (drawn as arrows) can represent motion

The length of the arrow is proportional to the magnitude

of the vector

y

x

Vector Addition

Vectors can slide around in the plane without changing, but...

Changing the magnitude changes the vector

Changing the direction changes the vector

To add two vectors graphically:

Slide them (without changing them) until they are “tip-to-tail”

The resultant vector is from the tail of the first to the tip of the second

Adding Perpendicular Vectors

To add perpendicular vectors

we can:

do it graphically or

use Pythagorean theorem and the tangent function

Let’s add two perpendicular

displacements, x and y, to get the resultant

displacement d

y

xx

x = 1.5 m right

y = 2.0 m down

d

tan = ——y

x

= tan–1(——)x

y

y

= tan–1(———)1.5 m

2.0 m

= 53

Resolving Vectors into Components

The vectors x and y are the

component vectors of the vector d

The x-component vector is always

parallel to the x-axis

The y-component vector is always

parallel to the y-axis

x

yd

We can resolve d back into its components using

the cos and sin functions:

dy

x

opphyp

adjadj

hyp

x

dcos = =

opp

hyp

y

dsin = =

x = d cos

y = d sin x = d cos = (2.5 m) cos (53°) = 1.5 m

y = d sin = (2.5 m) sin (53°) = 2.0 m

The components of d can be

represented by dx and dy

dx = x = 1.5 m

dy = y = -2.0 m

Notice dy is given a negative sign to

indicate that y is pointing down

Example of Components

Find the components of the

velocity of a helicopter

traveling 95 km/h at an angle of 35° to the ground

𝒗𝒙 = 𝟗𝟓 𝐜𝐨𝐬 𝟑𝟓° = 𝟕𝟕. 𝟖𝒌𝒎/𝒉

𝒗𝒚 = 𝟗𝟓 𝐬𝐢𝐧 𝟑𝟓° = 𝟓𝟒. 𝟓𝒌𝒎/𝒉

Add the following vectors

𝑨 = 𝟐. 𝟓𝒎/𝒔 𝜽 = 𝟒𝟓°

𝑩 = 𝟓. 𝟎𝒎/𝒔 𝜽 = 𝟐𝟕𝟎°

𝑪 = 𝟓. 𝟎𝒎/𝒔 𝜽 = 𝟑𝟑𝟎°

Hint: Vectors have both a magnitude and direction. θ is the direction.

𝑨𝒙 = 𝟐. 𝟓 𝐜𝐨𝐬 𝟒𝟓° = 𝟏. 𝟕𝟕𝒎/𝒔 𝑨𝒚 = 𝟐. 𝟓 𝐬𝐢𝐧𝟒𝟓° = 𝟏. 𝟕𝟕𝒎/𝒔

𝑩𝒙 = 𝟓. 𝟎 𝒄𝒐𝒔 𝟐𝟕𝟎° = 𝟎𝒎/𝒔 𝑩𝒚 = 𝟓. 𝟎 𝒔𝒊𝒏𝟐𝟕𝟎° = −𝟓. 𝟎𝟎𝒎/𝒔

𝑪𝒙 = 𝟓. 𝟎 𝒄𝒐𝒔 𝟑𝟑𝟎° = 𝟒. 𝟑𝟑𝒎/𝒔 𝑪𝒚 = 𝟓. 𝟎 𝒔𝒊𝒏𝟑𝟑𝟎° = −𝟐. 𝟓𝟎𝒎/𝒔

𝑹𝒙 = 𝑨𝒙 + 𝑩𝒙 + 𝑪𝒙 = 𝟔. 𝟏𝟎𝒎/𝒔 𝑹𝒚 = 𝑨𝒚 + 𝑩𝒚 + 𝑪𝒚 = −𝟓. 𝟕𝟑𝒎/𝒔

𝑹 = 𝑹𝒙𝟐 + 𝑹𝒚

𝟐 = 𝟖. 𝟑𝟕𝒎/𝒔

𝜽 = tan−𝟏𝑹𝒚

𝑹𝒙= 𝟑𝟏𝟕° 𝐨𝐫 − 𝟒𝟑. 𝟐°