2. 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

3. 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 For a vector-- Sign does not represent value, it represents direction! Traditionally: Up/Right (+) Down/Left (-) 3 m + (-4 m) = -1 m

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

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

6. Ø =30.0˚ d = 8.0 m dydy dxdx sinø = dyddyd d y = sinø(d) = sin30.0˚(8.0m) = 4.0 m cosø = dxddxd d x = cosø(d) = cos30.0˚(8.0m) = 6.9 m Adjacent Component! Opposite Component!

7. V Θ VxVx VyVy Adjacent component: V x = VcosΘ Opposite component: V y = VsinΘ tanΘ = VyVxVyVx Θ = tan -1 (V y /V x ) Be careful of the quandrant! V = √ V x 2 + V y 2

8. 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 d x = -22 m and d y = - 33 m. What is the magnitude and direction of this vector? What is the magnitude and direction of – B ?

9. 5) What are the components of a displacement of 12.0 m @ 230.0˚? 6) What is the resultant displacement when an object is moved 20.0 m to the east and 30.0 m to the south? 7) What are the components of a vector displacement of 17.0 m @ 125˚ ? 8) What is the resultant displacement of a box that is moved 4.00 m to the right and then 3.00 m upward?

10. Adding Vectors at Right Angles When adding vectors, use the head to tail method to get a proper resultant vector answer! A golfer on a flat green putts a ball 7.50 m in the East direction, and then putts the ball 2.50 m in the North direction into the hole. What single displacement would have provided him a single putt? A = 7.50 m, East B = 2.50 m, North

11. A B Head to Tail— On the head of the first goes the tail of the next vector! A = 7.50 m, East B = 2.50 m, North R = ? R R = A 2 + B 2 = 7.50 2 + 2.50 2 = 7.91 m ø tanØ = B / A = 2.50 / 7.50 =.333 Ø = tan -1 (.333) = 18.4˚

12. 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 North and drives another 33.0 km. What is her displacement? 4) What is the resultant displacement when a rock is moved 2.00 m to the right and then moved 3.50 m straight down?

13. 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:

14. In each case, the plane is heading (pointed in that direction) South, but… 288 m/s222 m/s257 m/s @ 277˚ Remember: Default reference frame is Earth!

15. 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?

16. V 1 = 12.0 m/s EastV 2 = 3.0 m/s North V1V1 V2V2 VRVR ø V R = (V 1 2 + V 2 2 ) = (12.0 2 + 3.00 2 ) = 12.4 m/s Ø = tan -1 (V 2 / V 1 ) = tan -1 (3.00/12.0) = 14.0˚ V R = 12.4 m/s @ 14.0˚ North of East

17. 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? 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?

18. 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?