Motion in 2 dimensions

3.1 -3.4

Vectors vs. Scalars

• Scalar- a quantity described by magnitude only. – Given by numbers and units only.– Ex. Distance, speed and mass

• Vector – a quantity described by both magnitude and direction– Numbers, units and direction (either words or

angles– Ex. Displacement, velocity, acceleration and

force

vectors

• Drawn with an arrow

• Length indicates magnitude

• Direction pointed

• Written in text either– As a boldfaced letter– Or as a letter with an arrow on top.

35 m East

Vector subtractionScalar multiplication

• Vectors may be subtracted by adding the opposite (or negative) of the second vector.

V2 – V1 = V2 + (– V1)

– This means that V1 is facing the opposite direction it was originally heading

• Multiplying a vector by a scalar simply magnifies its length if V= 3 m then 3V = 9 m

Vector addition

• Vectors may be added two ways:– One by graphing

• Vectors are drawn so that they remain in the same orientation but are placed tip to tail.

• Or by the parallelogram method

– One by adding the components of each and doing the Pythagorean theorem.

• The resultant vector is the outcome of adding 2 vectors together.

Parallelogram method:

• In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows:

• The resultant R is the diagonal of the parallelogram drawn from the common origin.

• R = resultant vectorPythagorean theorem

5 2 + 10 2 = R 2

R = 11.2 Km

Θ = tan -1 (5/10) =26.6° west of north

So sum = 11.2 Km @ 26.6° west of north

Component method – make a right triangle out of the individual vector

components

• The tip of the x-component vector is directly below the tip

of the original vector.

components

• , • .

When adding 2 vectors by the component method

• Find the x and y components of each vector• Organize in a table • Then add x component to x component and y

component to y component,

Vector X component Y component

A

B

Resultant X r Yr

• Use negative x components when vector is pointed to the left

• Use Negative y components pointed down

• Use the Pythagorean formula to find the length of the resulting vector.

• Find the angle using the Tan-1 (Yr /Xr)

• vector addition applet

• simulator

Example 1

• What is the vertical component of a 33 m vector that is at a 76° angle with the x axis?

• y –comp

• y = sin 76

33

y = 33 sin 76 = 32 m 76°

33 m

Example 2

• A plane is heading to a destination 1750 due north at 175 km/hr in a westward wind blowing 25 km/hr.

At what angle should the plane be oriented so that it reaches its destination?

• The wind will push the plane off course to the west by an angle of tan-1

(25/175) = 8° west of north so the plane needs to head 8° east of north.

Wind =25 km/hr

Plane = 175 km/hr

Example 3

• Using the same plane in example 2, what would the magnitude of the resultant vector be?– Since north and west, don’t need

components, just use the pythagorean theorem.

1752 + 25 2 = R 2

R = 177 km/hr

example 4

• What vector represents the displacement of a person who walks 15 km at 45° south of east then 30 km due west? • R = √( -19.4) 2 + (-10.6)2

• R = 22.1 km• Θ = tan -1 (-10.6/-19.4)• Θ = 29° south of west

15 km

30 km

Vector X – comp

Y -comp

15 km 10.6 -10.6

30 km -30 0

resultant -19.4 -10.6

45°