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  A x = A cos 

Vector Components  The y-component is found using the sin of the angle between the vector and the x-axis:  A y = 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 90 o 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 Motion  Projectile: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 45 0 angle  Same range results from any two angles that add up to 90 0  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

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