Vectors

Chapter 4

Vectors and Scalars Measured quantities can be of two types Scalar quantities: only require magnitude (and

proper unit) for description. Examples: distance, speed, mass, temperature, time

Vector quantities: require magnitude (with unit) and direction for complete description. Examples: displacement, velocity, acceleration, force, momentum

Representing Vectors

Arrows represent vector quantities, showing direction with length of arrow proportional to magnitude

In text, boldface type denotes vector When drawing, vectors can be moved on paper as

long as length and direction are not changed

Vector Addition The net effect of two or more vectors is another

vector called the resultant Vectors are not added like ordinary numbers,

directions must be taken into account For one-dimension motion, vector sum is same as

algebraic sum or difference For two dimensions, use graphical or mathematical

methods

Graphical Vector Addition

Involves using ruler and protractor to draw vectors to scale, measuring lengths and directions

Choose a suitable scale for the drawing Use a ruler to draw scaled magnitude and a

protractor for the direction

Graphical Vector Addition

Each successive vector is drawn with its tail at the arrowhead of the preceding vector

Resultant is vector from origin to end of final vector

Magnitude and direction can be measured Vectors can be added in any order without

changing the result

Vector Addition

Vector a plus vector b equals vector c Vector c is the resultant

a

bc

Vector Components Components of a vector are two or more vectors

that could be added together to equal the original vector

Vectors are resolved into right-angle components that are aligned with an x-y coordinate system

Using the angle between the vector and the x-axis ) the x-component is found using the cos of the angle

Ax = A cos

Vector Components

The y-component is found using the sin of the angle between the vector and the x-axis:

Ay = A sin

Vector Components

Algebraic Vector Addition Two vectors acting at right angles give a resultant

whose magnitude can be found using the Pythagorean theorem

Direction can be found using the tan-1 function If vectors act at angle other than 90o resolve

vectors into x and y components Add components to find components of resultant,

then add like right angle vectors

Other Vector Operations

Vector subtraction: the same as addition but with the reverse direction for the subtracted vector

Multiplying a vector by a scalar results in a vector in the same direction with a magnitude equal to the algebraic product

Projectile MotionProjectile:An object launched into the air

whose motion continues due to its own inertia Inertia: the tendency of a body to resist any

change in its motion Follows a parabolic path (trajectory)

Projectile Motion Constant vertical acceleration from gravity No horizontal acceleration, so horizontal

component of velocity is constant Horizontal and vertical motions are independent,

sharing only the time dimension

Velocity Vectors

Horizontal and Vertical Motion

Projectile Motion Horizontal distance of flight is called the range Range depends on launch angle and velocity Maximum range obtained from 450 angle Same range results from any two angles that

add up to 900 If launch velocity is enough so projectile path

matches earth’s curvature, it becomes satellite and orbits earth.

Solving Projectile Problems Separate vertical and

horizontal motions and work each separately.

Vertical motion is inde-pendent of horizontal motion

Gravity accelerates every-thing at the same rate whether it is moving sideways or not

Solving Projectile Problems

Solve one part of problem (usually vertical) for the time of flight and use this value to solve for distance in the other part.

Use constant acceleration equations for vertical problem, constant velocity for horizontal.

The Range of a Projectile: Horizontal Launch

Solve for time of free fall drop from vertical height:

Use time with initial velocity to find horizontal distance:

velocity vector components

g

yt

2

tvx x

The Range of a Projectile: Angle Launch

Resolve initial velocity into vertical and horizontal components

Find the time of flight in the vertical dimension Use positive sign for upward, negative for

downward User the time with the horizontal velocity

component to find the range