Graphical Vector Addition

Transcription

1 Vectors Chapter 4

2 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

3 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

4 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

5 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

6 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

7 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 (q), the x-component is found using the cos of the angle A x = A cosq

8 Vector Components The y-component is found using the sin of the angle between the vector and the x-axis: A y = A sin q

9 Vector Components

10 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

11 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

12 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) velocity vectors

13 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

14 Horizontal and Vertical Motion

15 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.

16 Solving Projectile Problems Separate vertical and horizontal motions and work each separately. Vertical motion is independent of horizontal motion Gravity accelerates everything at the same rate whether it is moving sideways or not

17 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.

18 The Range of a Projectile: Horizontal Launch Solve for time of free fall drop from vertical height: t y 2 g Use time with initial velocity to find horizontal distance: x v x t velocity vector components

19 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

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