Vectors

What’s a Vector? What’s a Vector?

�A vectorvectorvectorvector is a mathematical object

possessing, and fully described by, a

magnitudemagnitudemagnitudemagnitude and a directiondirectiondirectiondirection.

�The vector’s magnitude is equal to the length

of the arrow.

� its direction corresponds to where the arrow

is pointing

Vectors have direction

and magnitude

Vectors vs. Scalars Vectors vs. Scalars

� scalar quantity does not have a direction

�Examples of scalar quantities include the number

of words in this sentence and the mass of the

Hubble Space Telescope

Vectors can represent direction Vectors can represent direction Vectors can represent direction Vectors can represent direction

and magnitude of, e.g.,and magnitude of, e.g.,and magnitude of, e.g.,and magnitude of, e.g.,

�Velocity

� Acceleration

� Force

�Momentum

� Energy

Vector Example Vector Example Which of the following sentences deal with vector Quantities?

1. “I used to drive a 10-ton truck.”

2. “You’ll find a gas station if you follow this road 20 miles due north.”

3. “The 10-volt battery is the one on your left.”

(A) 1 only (B) 2 only (C) 3 only (D) 2 and 3 (E) 1, 2, and 3

It is the convention to print vectors in boldface type to

distinguish them from scalars. Graphically speaking, an arrow

can conveniently and effectively represent a vector. The length

of the arrow is proportional to its magnitude while the tip of the

arrow shows its direction

Vector Properties:

Vector Addition Vector Addition

�There are two equivalent ways to add vectors

graphically :

�Tip-to-Tail Method:

�Parallelogram Method :

Vector Addition , cont’dVector Addition , cont’d

Vector Example 2Vector Example 2

Vector Example 2Vector Example 2

Vector Example 2Vector Example 2

Pythagorean Theorem

In algebraic terms, a2 + b2 = c2 where c is the hypotenuse

while a and b are the legs of the triangle.

Example 1Example 1Example 1Example 1A motor boat traveling 4 m/s, East encounters a current traveling 3.0 m/s, North.a. What is the resultant velocity of the motor boat?b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?c. What distance downstream does the boat reach the opposite shore?

Example 2Example 2Example 2Example 2A motor boat traveling 4 m/s, East encounters a current traveling 7.0 m/s, North.a. What is the resultant velocity of the motor boat?b. If the width of the river is 80 meters wide, then how much time does it take the boat to travel shore to shore?c. What distance downstream does the boat reach the opposite shore?

ProjectilesProjectiles……

��are objects thrown or launched into the air and are subject are objects thrown or launched into the air and are subject

to gravityto gravity

��follow a curved path called a parabolafollow a curved path called a parabola

��are are a combination of una combination of un--accelerated xaccelerated x--direction motion direction motion

and freeand free--fallfall

��have have no acceleration in the xno acceleration in the x--directiondirection becausebecause velocity in velocity in

the xthe x--direction is constant.direction is constant.

��have acceleration in the yhave acceleration in the y--direction as gravity = 9.81 m/sdirection as gravity = 9.81 m/s22

Projectile Motion- Two DimensionalProjectile Motion- Two Dimensional

� To understand the motion a projectile

undergoes, examine the diagram on

the left.

The red ball is dropped at the same

time the yellow ball is launched

horizontally. Although it may seem

like the red ball would hit the ground

before the yellow one, if air

resistance is not taken into account,

both balls hit the ground at the same

time. The path of the yellow ball

represents the motion a projectile

undergoes.

Projectile Motion

Projectile Motion

Equations

� v =v =v =v = vvvvoooo+ at+ at+ at+ at

� x = xx = xx = xx = xoooo+ v+ v+ v+ v0000t + (1/2)att + (1/2)att + (1/2)att + (1/2)at2222

� vvvv2222 = v= v= v= voooo2222 + 2a(+ 2a(+ 2a(+ 2a(∆∆∆∆x)x)x)x)

Vertical Motion of a Projectile that Falls From Rest

� ∆∆∆∆y = y = y = y = ----(1/2) at(1/2) at(1/2) at(1/2) at2222

Horizontal Motion of a Projectile

� ∆∆∆∆x =x =x =x = vvvvxxxx((((∆∆∆∆t)t)t)t)

Projectile Motion