1. Physics Lesson 5 Two Dimensional Motion and Vectors Eleanor Roosevelt High School Mr. Chin-Sung Lin

2. Two Dimensional Motion and Vectors  Scalars & Vectors  Vector Representation  One-Dimensional Vector Addition  Two-Dimensional Vector Addition  Vector Resolution  Vector Addition Through Resolution  Vector Application: Relative Velocity

3. Scalars & Vectors

4. Comparison of Scalars & Vectors ScalarsVectors Magnitude Direction Physical Quantities

5. Comparison of Scalars & Vectors ScalarsVectors Magnitude Direction Physical Quantities 3 m/s 60 o Scalars & Vectors

6. Examples of Scalars & Vectors ScalarsVectors Physical Quantities Scalars & Vectors displacement velocity acceleration force distance speed acceleration mass

7. Vector Representation

8. Arrows  An arrow is used to represent the magnitude and direction of a vector quantity  Magnitude: the length of the arrow  Direction: the direction of the arrow Head Tail Magnitude Direction Vector Representation

9. Equality of Vectors  Vectors are equal when they have the same magnitude and direction, irrespective of their point of origin Magnitude Direction Vector Representation

10. Negative Vectors  A vector having the same magnitude but opposite direction to a vector A Vector Representation - A

11. One-Dimensional Vector Addition

12. Vector Addition (Same Direction)  The result of adding two vectors (resultant) with the same direction is the sum of the two magnitudes and the same direction One-Dimensional Vector Addition 10 m 5 m

13. Vector Addition (Opposite Directions)  The result of adding two vectors (resultant) with opposite directions is the difference of the two magnitudes and the direction of the longer one One-Dimensional Vector Addition 10 m -5 m 5 m

14. Two-Dimensional Vector Addition

15. Vector Addition (Parallelogram Method)  The resultant is the diagonal of the parallelogram described by the two vectors Two-Dimensional Vector Addition Resultant B A

16. Vector Addition (Head-Tail Method)  Many vectors can be added together by drawing the successive vectors in a head-to-tail fashion. The resultant is from the tail of the first vector to the head of the last vector Two-Dimensional Vector Addition Resultant B A

17. Vector Subtraction  One vector subtracts another vector is the same as one vector adds another negative vector Two-Dimensional Vector Addition A A – B = A + (-B) B

18. Vector Subtraction  One vector subtracts another vector is the same as one vector adds another negative vector Two-Dimensional Vector Addition Resultant - B A A – B = A + (-B)

19. Vector Resolution

20. Component Vectors  Any vector can be resolved into two component vectors (vertical and horizontal components) at right angle to each other Vector Resolution Horizontal component Vector Vertical componen t

21. Component Vectors  The process of determining the components of a vector is called vector resolution Vector Resolution Horizontal component Vector Vertical componen t

22. Calculate Component Vectors  The magnitude of the horizontal component v x = v cos θ  The magnitude of the vertical component v y = v sin θ Vector Resolution V x = V cos θ V V y = V sin θ θ

23. Two-dimensional vector addition through vector resolution

24. Two-Dimensional Vectors Addition  Resolve vectors into horizontal and vertical components  Add all the horizontal components of the vectors  Add all the vertical components of the vectors.  Find the final resultant by adding the horizontal and vertical components of the final resultant Vector Addition through Resolution

25. Two-Dimensional Vectors Addition Vector Addition through Resolution A x A AyAy ByBy R BxBx B R x R y

26. Two-Dimensional Vectors Addition Vector Addition through Resolution 34.6 m/s 40.0 m/s 20.0 m/s -26.0 m/s -15.0 m/s 30.0 m/s 19.6 m/s -6.0 m/s 30 o 60 o 20.5 m/s -16.9 o 34.6 m/s – 15.0 m/s = 19.6 m/s 20.0 m/s – 26.0 m/s = -6.0 m/s tan -1 (-6.0 m/s /19.6 m/s) = -16.9 o sqrt (19.6 2 + 6.0 2 ) m/s = 20.5 m/s -30.0 m/s sin (60 o ) = -26.0 m/s -30.0 m/s cos (60 o ) = -15.0 m/s 40.0 m/s sin (30 o ) = 20.0 m/s 40.0 m/s cos (30 o ) = 34.6 m/s

27. Vector Application: Relative Motion

28. Relative Velocity  Relative velocity is the vector difference between the velocities of two objects in the same coordinate system Vector Application

29. Relative Velocity  For example, if the velocities of particles A and B are v A and v B respectively in the same coordinate system, then the relative velocity of A with respect to B (also called the velocity of A relative to B) is v A – v B VAVA V A – V B VBVB Vector Application

30. Relative Velocity  The relative velocity vector calculation for both one- and two-dimensional motion are similar  The velocity vector subtraction (v A – v B ) can be viewed as vector addition (v A + (–v B )) Vector Application VAVA V A +(–V B ) -V B

31. Relative Velocity  Conversely the velocity of B relative to A is v B – v A Vector Application VAVA V B – V A VBVB

32. Q & A

33. The End