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Ch. 3: Kinematics in 2 or 3 Dimensions; Vectors

Ch. 3: Kinematics in 2 or 3 Dimensions; Vectors. Outline. Two Dimensional Vectors Magnitude & Direction Algebraic Vector Operations Equality of vectors Vector addition Multiplication of vectors with scalars Scalar product of two vectors (a later chapter!) Vector product of two vectors

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Ch. 3: Kinematics in 2 or 3 Dimensions; Vectors

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  1. Ch. 3: Kinematics in 2 or 3 Dimensions; Vectors

  2. Outline Two Dimensional Vectors Magnitude & Direction Algebraic Vector Operations Equality of vectors Vector addition Multiplication of vectors with scalars Scalar product of two vectors (a later chapter!) Vector product of two vectors (a later chapter!)

  3. Vectors • General discussion. Vector A quantity with magnitude & direction. Scalar A quantity with magnitude only. • Here: We’ll mainly deal with Displacement  D & Velocity  v But, our discussion will be valid for any kind of vector! • This chapter has a lot of math! Understanding it requires a detailed knowledge of trigonometry. Problem Solving A diagram or sketch is helpful & vital! I don’t see how it is possible to solve a vector problem without a diagram!

  4. Rectangular or Cartesian Coordinates Review of “standard” coordinate axes. A point in the x-y plane is labeled (x,y) Note, if it is convenient, we could reverse + & - Coordinate Systems - ,+ +,+ - , - + , - Standard Rectangular (x-y) Coordinate Axes

  5. Vector & Scalar Quantities Vector  A quantity with magnitude & direction. Scalar  A quantity with magnitude only.

  6. Equality of Two Vectors For 2 vectors, A & B, A = B means that A & B have the same magnitude & direction.

  7. Sect. 3-2: Vector Addition, Graphical Method • Addition of scalars is “Normal” arithmetic! • Addition of vectors is not so simple! • For 2 vectors in the same direction: • We can also use simple arithmetic Example:Travel 8 km East on day 1, 6 km East on day 2. Displacement = 8 km + 6 km = 14 km East Example:Travel 8 km East on day 1, 6 km West on day 2. Displacement = 8 km - 6 km = 2 km East “Resultant” = Displacement

  8. Adding 2 Vectors in the Same Direction:

  9. Graphical Method • For 2 vectors NOTalong the same line, adding is more complicated: Example:D1= 10 km East, D2 = 5 km North. What is the resultant (final) displacement? • There are 2 Methods of Vector Addition: • Graphical (there are 2 methods of this also!) • Analytical (TRIGONOMETRY)

  10. In this special case ONLY, D1 is perpendicular to D2. So, we can use the Pythagorean Theorem. • For 2 vectors NOT along same line: D1 = 10 km E, D2 = 5 km N. We want to find the Resultant = DR = D1 + D2 = ? = 11.2 km Note! DR< D1 + D2 (scalar addition) The Graphical Method requires measuring the length of DR & the angle θ. Do that & find DR = 11.2 km, θ = 27º N of E

  11. This example illustrates the general rules(for the “tail-to-tip” method of graphical addition).Consider R = A + B: 1. Draw A & Bto scale. 2. Place the tail of B at the tip of A 3. Draw an arrow from the tail of Ato the tip of B That arrow is the ResultantR (measure the length & the angle it makes with the x-axis)

  12. Order isn’t important!Adding vectors in the opposite order gives the same result. In the example,DR = D1 + D2 = D2 +D1

  13. Graphical Method Continued Adding 3 (or more) vectors V = V1 + V2 + V3 Even if the vectors are not at right angles, they can be added graphically by using the tail-to-tip method.

  14. A second graphical method of adding vectors: (100% equivalent to the tail-to-tip method!) V = V1 + V2 1. Draw V1& V2to scale from common origin. 2.Construct a parallelogram with V1& V2 as 2 of the 4 sides. Then, theResultant V = The diagonal of the parallelogram from the common origin (measure the length and the angle it makes with the x-axis)

  15. So, The Parallelogram Method may also be used for the graphical addition of vectors. A common error! Mathematically, we can move vectors around (preserving magnitudes & directions)

  16. Sect. 3-3: Subtraction of Vectors • First, Definethe Negative of a Vector: -V the vector with the same magnitude (size) as Vbut with the opposite direction. V + (- V)  0 Then, to subtract 2 vectors, add one vector to the negative of the other. • For 2 vectors, V1 & V2: V1 - V2  V1 + (-V2)

  17. Multiplication by a Scalar A vector V can be multiplied by a scalar c V' =cV V' vector with magnitudecVthe same direction asV If c is negative, the result is in the opposite direction.

  18. Example • A two part car trip: First, displacement:A = 20 kmdue North. Then, displacementB = 35 km 60º West of North. Find (graphically) the resultant displacement vectorR(magnitude & direction). R = A + B Use aruler & protractorto findthe length of R & the angle β: Length = 48.2 km β = 38.9º

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