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Chapter 4 Vectors

Chapter 4 Vectors. DEHS 2011-12 Physics 1. What is (and is not) a vector?. A scalar is a quantity that is described ONLY by magnitude (a number with units). It can be positive, negative or zero Ex: distance, time, speed

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Chapter 4 Vectors

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  1. Chapter 4Vectors DEHS 2011-12 Physics 1

  2. What is (and is not) a vector? • A scalar is a quantity that is described ONLY by magnitude (a number with units). It can be positive, negative or zero • Ex: distance, time, speed • A vector is a quantity that is described with BOTH magnitude and direction. It can be zero or nonzero • Ex: position, displacement, velocity, acceleration, forces

  3. Problem: Where is the library? • The library’s location is it’s position, which is a vector quantity • Say you only knew how far (distance, a scalar) the library was from you, a distance of 0.5 mi. It could lie anywhere on a circle of radius 0.5 mi

  4. Representing a Vector • It is useful to represent vectors visually (graphically) by ARROWS • Direction of the vector is indicated by the direction of the arrow • Magnitude is indicated by the length of the arrow (longer arrow = greater magnitude) • Arrows in the same direction with the same length (same magnitude) represent equal vectors • You can translate (draw in different places) vectors and still keep them the same, just don’t change the length or direction

  5. Identical vectors are different locations

  6. Describing a Vector • Describing magnitude: you must express the magnitude of a vector with a number and units • Describing direction: you must give an angle AND what axis the angle is measured from

  7. Some Familiar Properties of Vectors • You can: • Add vectors (you get a vector) • Subtract vectors (you get a vector) • They follow: • The commutative property (order of addition doesn’t matter) • The identity property (adding zero doesn’t change it)

  8. Some Familiar Properties of Vectors • Changing the order of subtraction gives you the opposite • Solving addition/subtraction of vectors is the same as with scalars

  9. Some Unfamiliar Properties of Vectors • There are two separate processes for multiplying two vectors (will NOT be covered) • Multiplying a scalar and a vector will give you a vector

  10. Example 4-1 Find B for the following values of A and a, where A = 4.0 m/s @ 30° N of W and a = 2.0 A = 18 m @ 50° above the −x-axis and a = −0.5 A = 6.0 m @ due west and a = −1.0

  11. Scalar Components of a Vector • The scalar component of a vector are the projections of the vector onto the x & y axis • They can be positive, negative, or zero • Notation: Ax and Ay are the scalar components of vector A • θ MUST be measured from an x-axis • The signs of the components depends on what quadrant the vector is in

  12. Example 4-2 A man wants to find the height of a cliff. He stands with his back to the base of the cliff then marches straight away from it for 500 ft. At this point he lies on the ground and measures the angle from the horizontal to the top of the cliff. If the angle is 34° (a) what is the straight-line distance from the man to the the top of the cliff? (b) what is the height of the cliff?

  13. Example 4-2

  14. Reconstructing a Vector from its components Since the components of a vector (Ax& Ay) form the legs of a right triangle, where the vector is the hypotenuse, the vector’s magnitude is calculated with the Pythagorean Theorem

  15. Reconstructing a Vector from its components Since the components of a vector (Ax& Ay) form the legs of a right triangle, the vector’s angle with an X-AXIS is calculated using the inverse tangent, and the axis reference is found by looking at the signs of the components Ax> 0 Ax> 0 angle is described as Ax< 0 Ax> 0 angle is described as Ax< 0 Ax< 0 angle is described as Ax> 0 Ax< 0 angle is described as

  16. Example 4-3 Given that Ax= −9.0 m and Ay= 40 m, find A in magnitude/direction notation Given that Bx= −8.0 m and By= 15 m, find B in magnitude/direction notation

  17. Adding Vectors(Graphical Method) • Head to tail method: to add the vectors A and B, place the tail of B at the tip of A. The sum, C = A + B, is the vector extending from the tail of A to the head of B • To add more than two vectors, keep placing them head to tail and draw a vector from the tail of the first vector to the head of the last vector

  18. Example 4-4 Given that A = 5.00 m @ 60.0° above the +x-axis and B = 4.00 m @ 20.0° above the +x-axis. Use the graphical method of vector addition to find vector C, where C = A + B (you will need a protractor!)

  19. Subtracting Vectors(Graphical Method) • Subtracting a vector is the same as adding its opposite • The opposite of a vector is represented graphically by an arrow of the same length as the original vector, but pointing in the opposite direction becomes...

  20. Adding/Subtracting Vectors(Component Method) • Resolve each vector into its components using sine and cosine. • SET UP A TABLE! Add or subtract the components following the vector equation. • Express the answer in desired notation (either unit vector or magnitude/direction) • If you are expressing your answer in magnitude/direction notation, it will require two more steps

  21. Example 4-5 Taking a walk around your backyard. You take 5 paces due North, 3 paces due East, 4 paces to 45° S of W. Find your total displacement D using the component method, expressing your answer in magnitude/direction notation.

  22. Example 4-6

  23. Unit Vectors • Unit vectors are dimensionless vectors of unit magnitude pointing in the positive direction • Notation: = “x hat”, = “y hat”

  24. Unit Vector Notation • We can use our rule for adding vectors multiplying vectors by scalars to express vectors as: • With unit vector notation, addition and subtraction are straightforward:

  25. Position & Displacement Vectors • The position vector is defined as: • The displacement vector is defined as:

  26. Average Velocity Vector • Since is a vector, must also be a vector as it is times the scalar (1/Δt) • It is in the same direction as • It will be shorter than as long as t > 1 s

  27. Example4-7 A dragonfly is observed initially at the position Three seconds later it is at the position What is the dragonfly’s displacement and average velocity during this time?

  28. Example4-8 Find the speed and direction of motion for a rainbow trout whose velocity is

  29. Average Acceleration Vector Consider a car traveling in a circular path at a speed of 12 m/s, calculate the average acceleration of the car if it takes 10 s to complete ¼ of a revolution.

  30. Instantaneous v and a vectors • If an object’s path is drawn, the velocity vector v is always in the direction of the object’s motion and will appear to be tangent to the object’s path • The acceleration vector a can point in directions other that the direction of motion, and in general it does • a is ALWAYS in the direction of Δv • The angle between the a and v vectors will determine if the speed and/or the direction of motion will change (summarized on the next slide)

  31. What is the effect on the object’s speed and direction of motion at points 1, 2, 3, and 4?

  32. Relative Motion • For an object that is within a moving frame of reference, you must consider motion relative to different frames of reference • Ex: You are driving North on the highway at 55 mph as you pass a car that is going 50 mph. How fast and in what direction does the other car appear to be moving to you? How fast do you appear to be moving to the other driver?

  33. Generalized Relative Motion Eqns • Reversing the subscripts reverses the direction of the velocity: • The general equation is where 1 & 3 are the objects and 2 can be anything

  34. Example4-9 A person climbs up a ladder on a moving train with a velocity of 3 m/s relative to the train. If the train moves relative to the ground with a velocity of 5 m/s, what is the velocity of the person relative to an observer watching from the ground?

  35. Example4-10 You are riding in a boat whose speed relative to the water is 6.1 m/s. The boat points at an angle of 25° upstream on a river flowing 1.4 m/s. (a) What is your velocity relative to the ground? (b) Suppose the speed of the boat relative to the water remains the same, but the direction in which it points is changed. What angle is required for the boat to go straight across the stream? What is the boat’s speed relative to the ground be?

  36. Example4-10

  37. Example4-11 A couple is shopping on the 2nd floor of the mall. The wife wants to go to the 1st floor and the husband wants to the 3rd floor. The both hop on an escalator. The escalators moves them both along at 2 m/s. Both escalators make an angle of 45° with the horizontal. Find the husband’s velocity relative to his wife vhusband 45° 45° vwife

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