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VECTOR ADDITION

VECTOR ADDITION. ACP Physics. Vectors. Vectors Quantities have magnitude and direction and can be represented with; Arrows Sign Conventions (1-Dimension) Angles and Definite Directions (N, S, E, W). Vectors. Every Vector has two parts. A Head and a Tail.

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VECTOR ADDITION

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  1. VECTOR ADDITION ACP Physics

  2. Vectors Vectors Quantities have magnitude and direction and can be represented with; • Arrows • Sign Conventions (1-Dimension) • Angles and Definite Directions (N, S, E, W)

  3. Vectors • Every Vector has two parts. • A Head and a Tail. ------------------ Tail Head

  4. Vectors • One dimensional Vectors can use a “+” or “–” sign to show direction. + 50 m/s a- 45 m b - 9.8 m/s2i+ 60 N h

  5. Picking an Angle for 2-Dimensional Vectors • Pick the angle from the tail of the Vector. • Find a Definite Direction w.r.t. the Vector (N, S, E, W). • Write your angle w.r.t. the Definite Direction.

  6. Addition of Scalars • When adding scalars, direction does not matter so we add or subtract magnitudes. • 50 kg + 23 kg = 73 kg • 25 s – 13 s = 12 s

  7. Addition of Vectors • Adding vectors is more complicated because the direction affects how the vectors can be combined. A aBlA d B m A + B = ? A + B = ?

  8. Addition of 1-Dimensional Vectors One Dimensional Vectors are either in the same or opposite directions. • If the Vectors are in the same direction we add their magnitudes. • If the Vectors are in opposite directions we subtract their magnitudes.

  9. Addition of 2-Dimensional Vectors Addition of 2-Dimensional Vectors requires other methods. • Graphical Method • Mathematical or Triangle Method • Component Method (Best Method)

  10. Properties of Vectors • The addition of vectors yields a RESULTANT (R) R = A + B • Order of addition does not matter! A + B = B + A = R A+B+C = C+B+A = B+C+A = R

  11. Properties of Vectors • Vectors that are Perpendicular to each other are INDEPENDENT of each other. • Ex. Boat traveling perpendicular to the current. • Ex. Projectile thrown in the air.

  12. Graphical Addition of VectorsR = A + B Adding vectors Graphically has certain advantages and disadvantages. • Not a mathematical method

  13. Advantages You can add as many vectors as you want Importance: Vector Diagram Disadvantages Need a Ruler and a Protractor Not the most accurate method Graphical Addition of VectorsR = A + B

  14. Graphical Addition of VectorsR = A + B • Pick an appropriate scale to draw vectors • Draw the First Vector (A) to scale • Draw the Next Vector (B) from the Head of the Previous Vector (A)

  15. Graphical Addition of VectorsR = A + B • Draw the Resultant (R) from the Tail of the First to the Head of the Last • Measure the Resultant (R), length and direction, and use your scale to give your answer

  16. Vectors are Relative • Vectors that are measured depend on their reference point or frame of reference • Velocities measured are relative w.r.t. the Observer • 1-Dimensional (Ex. Bus Ride) • 2-Dimensional (Ex. River Crossing)

  17. Subtraction of Vectors R = A – B • Subtraction is the Addition of a Negative Vector A - B = A + (-B) • So what does –B mean? • -B is the vector that has the same magnitude of B but in the opposite direction!

  18. Subtraction of Vectors R = A – B • Ex. A = 50 N a B = 40 N h-B = 40 N , • A + B = ? A – B = ? • Does (A – B)= (B – A) ??? • No!!! A – B = -B + A = -(B – A)

  19. Advantages Mathematical Method (More Accurate) Importance: Working with Right Triangles Disadvantages Can only add two vectors at a time Vector Addition: Triangle Method

  20. Vector Addition: Triangle Method 1. Use a Vector Diagram to make your triangle (see Graphical Method) 2. See what kind of Triangle you have

  21. Vector Addition: Triangle Method 3. If you have a Right Triangle use: • Pythagorean Theorem • c2 = a2 + b2 • Tangent Function • tan q = opposite side = O adjacent side A

  22. Vector Addition: Triangle Method • If you do not have a Right Triangle use: • Law of Cosines • c2 = a2 + b2 – 2abCos(C) • Law of Sines • Sin(A) = Sin(B) = Sin(C) a b c

  23. Perpendicular Components of a Vector Every Vector can be split into two perpendicular components • A = Ax + Ay • Ax = Horizontal component of A • Ay = Vertical component of A

  24. Perpendicular Components of a Vector Ax and Ay are one dimensional vectors • Ax - a“+” d“-” • Ay - h “+”i “-”

  25. To Find Perpendicular Components of a Vector • Since the components(Ax, Ay) are perpendicular, they form a right triangle with. Vector A • Use the sine and cosine functions to find Ax and Ay

  26. To Find Perpendicular Components of a Vector • sin q = Opposite = O Hypotenuse H • cos q = Adjacent = A Hypotenuse H

  27. Component Method Advantages to the Component Method • Do not need a ruler or a protractor (More accurate) • Can add multiple vectors • Every triangle is a right triangle!!

  28. Component Method • Break up each vector into its perpendicular components (Ax, Ay, Bx, By, Cx, Cy, etc,) • Add up all the x-components to get the x-component of Resultant (Rx = Ax + Bx + Cx + … etc)

  29. Component Method • Add up all the y-components to get the y-component of Resultant (Ry = Ay + By + Cy + … etc) • Recombine Rx and Ryto make the Resultant, R (R = Rx + Ry) (Use the Triangle Method for Right Triangles) • Every triangle is a Right Triangle

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