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Vectors and Relative Motion

Vectors and Relative Motion. Vector Quantity Fully described by both magnitude (number plus units) AND direction Represented by arrows -velocity -acceleration -force. Scalar Quantity Fully described by magnitude (number plus units) alone -mass -temperature. Adding Vectors.

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Vectors and Relative Motion

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  1. Vectors and Relative Motion Vector Quantity Fully described by both magnitude (number plus units) AND direction Represented by arrows -velocity -acceleration -force Scalar Quantity Fully described by magnitude (number plus units) alone -mass -temperature

  2. Adding Vectors Vectors in one dimension are added algebraically: 3 m, North + 4 m, North = 7 m, North 3 m + 4 m = 7 m 3 m, North + 4 m, South = 1 m, South 3 m + (-4 m) = -1 m For a vector-- Sign does not represent value, it represents direction! Traditionally: Up/Right (+) Down/Left (-)

  3. Adding Vectors in 2 Dimensions- Vectors add Trigonometrically Using Head to Tail Method: 3.0 m + 4.0 m = 2.2 m 3.0 m + 4.0 m = 6.5 m 4.0 m 3.0 m 4.0 m N 6.5 m 2.2 m 3.0 m Vector diagrams show magnitude and direction of vectors and their resultant! 8.0 N + 6.0 N = ? 2.0 N ≤ ? ≤ 14 N

  4. Notice Vector Direction: In relation to + x axis

  5. Vector Direction: By agreement, vectors are generally described by how many degrees the vector is rotated from the + x axis 30˚ 30˚ 150˚ Negative 2D vectors: A - A 180˚ opposite

  6. Resolution (Decomposition) of Vectors If you move a box 8.0 m @ 30.0˚ from O: By Geometry: 8.0 m 4.0 m 30.0˚ 6.9 m The box has moved– 6.9 m to the right ( +x) And 4.0 mupward (+y) These values would be the components of the given vector !

  7. d = 8.0 m dy Ø =30.0˚ dx dy d sinø = dy = sinø(d) = sin30.0˚(8.0m) = 4.0 m Opposite Component! dx d dx = cosø(d) = cos30.0˚(8.0m) cosø = = 6.9 m Adjacent Component!

  8. V V = √ Vx2 + Vy2 Vy Θ Vx Adjacent component: Vx = VcosΘ Opposite component: Vy = VsinΘ Vy Vx Θ = tan-1 (Vy/Vx) tanΘ = Be careful of the quandrant!

  9. 1) A man walks 5.0 km to the East and then walks 3.0 km to the North. What is his displacement from where he started? 2) What are the components of a vector displacement of 12.0 m @ 32.0˚? 3) If a student walks 56.0 m North and then turns West and walks another 85.0 m, what is his displacement? 4) Vector B has components of dx = -22 m and dy = - 33 m. What is the magnitude and direction of this vector? What is the magnitude and direction of – B ?

  10. 5) What is the resultant displacement when a box is moved 5.00 m in the x direction and then -7.50 m in the y direction? 6) What are the components of the vector shown below? A = 27.3 m ø Ø = 32.8˚ A

  11. Adding Vectors Using Components When adding two (or more) vectors, adding the components will give the components of the Resultant vector: A golfer on a flat green putts a ball 7.50 m in the Northeast direction, but misses the hole. He then putts the ball 2.30 m @ 38.0˚ South of straight East and sinks the putt for a bogey. What single putt would have saved par? d1 = 7.50 m @ 45.0˚ d2 = 2.30 m @ - 38.0˚

  12. -38.0˚ Head to Tail—On the head of the first goes the tail of the next vector! d1 d2 45.0˚ d1x = cos45.0˚(7.50 m) = 5.30 m d1y = sin45.0˚(7.50 m) = 5.30 m d2x = cos(-38.0˚)(2.30 m) = 1.81 m d2y = sin(-38.0˚)(2.30 m) = -1.42 m

  13. dx = d1x + d2x = 5.30 m + 1.81 m = 7.11 m dy = d1y + d2y = 5.30 m + (- 1.42 m) = 3.88 m d = √ dx2 + dy2 = √(7.11)2 + (3.88)2 = 8.10 m Ø = tan-1 (dy / dx) = tan-1(3.88 / 7.11) = 28.6˚ d2 The single (resultant) putt: d1 d ø d = 8.10 m @ 28.6˚

  14. 1) What is the resulting displacement when an object is moved 10.0 m to the North and then 5.0 m to the east? 2) A man leaves his house and walks 6.00 km to the West and then turns and walks 3.50 km to the South. What is his displacement? 3) A woman drives straight East for 65.0 km and then turns 30.0˚ North of East and drives another 33.0 km. What is her displacement? 4) A = 25.0 N @ 33.0˚ B = 57.7 N @ 152˚ Find the resultant when vector A is added to vector B.

  15. 5) Add the following three vectors: A = 225 m α = 28.0˚ A B = 275 m Β = 56.0˚ B α β ø C = 325 m ø = 15.0˚ C

  16. Relative Velocity Velocities are vectors and add like vectors: A plane flies through the air at a speed of 255 m/s. The air speed is 33.0 m/s. The velocity of the plane relative to the ground depends upon direction:

  17. In each case, the plane is heading (pointed in that direction) South, but… 288 m/s 222 m/s 257 m/s @ 277˚ Remember: Default reference frame is Earth!

  18. A boat travels at 12.0 m/s relative to the water and heads East across a river that flows North at 3.00 m/s. What is the speed and direction of the boat relative to the shore?

  19. Vbw = 12.0 m/s @ 0˚ Vwg = 3.0 m/s @ 90.0˚ Vbg Vwg ø Vbw Vbg = (V12 + V22) = (12.02 + 3.002) = 12.4 m/s Ø = tan-1(V2 / V1) = tan-1(3.00/12.0) = 14.0˚ Vbg = 12.4 m/s @ 14.0˚

  20. 1) A boat heads West across a stream that flows South. What is the velocity of the boat relative to the shore if it heads across with a speed of 8.3 m/s while the water flows South at 2.4 m/s? 2) An airplane heading due North at 325 m/s encounters a wind of 55.0m/s from the East. What will the velocity of the plane be now? 3) A barge heading West down a still river travels at 5.0 m/s. A man walks across the barge from North to South at 2.0 m/s. What is the velocity of the man as viewed from a bridge above?

  21. 4) A boat wants to travel directly across a river that flows South at 3.0 m/s. If the boat travels at 7.0 m/s in still water, what heading must it take to go straight across? With what speed will the boat travel straight across? 5) An airplane has a velocity of 285 m/s @ 215˚ while flying through a crosswind. What is the heading of the plane? What is the velocity of the wind?

  22. 6) A man in a blue car traveling at 25.0 m/s @ 25.0˚ views a second red auto traveling at 32.0 m/s @ 215˚. What is the velocity of the red car relative the the man in the blue car? 7) An airplane flies at 225 m/s @ 45.0˚ North of east. A second plane flies at 175 m/s @ 35.0˚ South of North. What is the velocity of the the first plane relative to the second? What is the velocity of the second plane relative to the first?

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