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