Physics 1710 Chapter 3 Vectors

1. 0 Physics 1710 Chapter 3 Vectors Demonstration: Egg Toss REVIEW

2. No Talking! Think! Confer! 0 Physics 1710 Chapter 3 Vectors Why did the egg not break the first time it was caught but did the second time? REVIEW

3. 0 Physics 1710 Chapter 3 Vectors Different acceleration a = (vfinal2 – v initial 2)/ (2∆x) Why wear a seat belt or use air bags? REVIEW

4. 0 Physics 1710 Chapter 3 Vectors Seat belt: Air Bag Video

5. 0 Physics 1710 Chapter 3 Vectors 1′ Lecture: A Vector is a quantity that requires two or more numbers to define it and acts like the displacement vector. The magnitude of a vector is the square root of the sum of the squares of its components. A vector makes an angle to the x-axis whose tangent is equal to the ratio of the y-component to the x-component.

6. 0 Physics 1710 Chapter 3 Vectors Stranded Motorist asks horse cart driver, “Is it far to Budapest?” “Nem! It is not far.” Is it far to Budapest?

7. 0 Physics 1710 Chapter 3 Vectors “Then, may I have a ride?” “Egan! Climb up.” After a long time the Motorist says, “I thought you said it was not far.” What is the problem? The difference between distance and displacement. The driver replies, “Oh! Now it is veryfar to Budapest.”

8. 0 Physics 1710 Chapter 3 Vectors Turn to your classmate and the one in the odd numbered seat, tell the other where is the Student Union. Where is the Student Union? Position is a vector.

9. 0 Physics 1710 Chapter 3 Vectors AScalaris a entity that requires only one number to characterize it fully. (Like a scale.) Examples: What time is it? What is your weight? What is the temperature of the room? What is the weight of 100. Kg man? Weight = g m = 9.80 N/kg (100. kg) = 980 N.

10. 0 Physics 1710 Chapter 3 Vectors A vector is a quantity that requires more than one “component” to “tell the whole story. Example: Where is the treasure buried in the field? Use “orthogonal,” that is, perpendicular axes.

11. 4th St and 2nd Ave 2nd St and 4th Ave 0 Physics 1710 Chapter 3 Vectors (4,2) Location in Manhattan (2,4)

12. 0 Physics 1710 Chapter 3 Vectors Position in 2-Dimensions or higher is a VECTOR. We use boldface, not italic, to denote a vector quantity, italics to denote the scalar components. We often represent a vector as a position on a graph with an arrow connecting the origin to the position.

13. θ j i Physics 1710Chapter 2 Motion in One Dimension—II 0 2-Dimensional Vector Position Vector r r = (x,y) = x i + y j x= r cos θ y = r sin θ | r | = r = √(x 2 + y 2), θ = tan –1(y/x) Y(m) x r y X (m)

14. 0 Physics 1710 Chapter 3 Vectors 80/20 Fact: The length of the arrow represents the magnitude of the vector. In orthogonal coordinates, the magnitude of vector A given by: ∣A∣ = √ [Ax2 + Ay2 + Az2 ]

15. 0 Physics 1710 Chapter 3 Vectors 80/20 Fact: The direction of the vector A is characterized (two dimensions) by the angle  it makes with the “x-axis.” tan θ= Ay / Ax

16. Physics 1710Chapter 2 Motion in One Dimension—II 0 2-Dimensional Vector Position Vector r | r | = r = √(x 2 + y 2) = √(2.0 2 + 1.5 2) = √(4.0+ 2.25 ) = √(6.25) = 2.5 m Y(m) x r y X (m)

17. 0 80/20 Fact: Physics 1710 Chapter 3 Vectors One may combine vectors by “ vector addition”: C = A + B Then C x= Ax+ Bx & Cy=Ay+ By Key point: Add the components separately. Observe strict segregation of x and y parts.

18. 0 80/20 Fact: Physics 1710 Chapter 3 Vectors The product of a scalar and a vector is a vector for which every component is multiplied by the scalar: C = k A Cx = k Ax Cy = k Ay Cz = k Az

19. 0 Physics 1710 Chapter 3 Vectors N.B. ( Note Well): ⅠA + BⅠ ≠ (A + B)

20. 0 Physics 1710 Chapter 3 Vectors Note: ⅠA + BⅠ = √[(Ax+ Bx ) 2 + (Ay+ By ) 2 ] ≤ (A + B) Proof: (Ax+ Bx ) 2 + (Ay+ By ) 2≤ (A+B)2 = A2 +2AB +B2 LHS = Ax2 +Ay2 + Bx 2 + By 2 + 2AxBx + 2AyBy RHS = Ax2 + Ay2+ Bx 2 + By 2 +2√(Ax2 Bx2 + Ay2 By2 + Ay2 Bx2 + Ax2 By2) LHS ≤ RHS 2AxBx + 2AyBy≤ 2√(Ax2 Bx2 + Ay2 By2 + Ay2 Bx2 + Ax2 By2) Ax 2Bx 2+ 2 AxBx AyBy +Ay2By 2≤ Ax2 Bx2 + Ay2 By2 + Ay2 Bx2 + Ax2 By2 2 AxBx AyBy ≤ Ay2 Bx2 + Ax2 By2 iff 0 ≤ (AyBx- AxBy) 2

21. 0 80/20 Fact: Physics 1710 Chapter 3 Vectors We often designate the components of the vector by unit vectors ( i, j, k ) the x,y, and z components, respectively. Thus, 2.0 i + 3.0 j has an x-component of 2.0 units and a y-component of 3.0 units. Or (2.0, 3.0)

22. 0 Physics 1710 Chapter 3 Vectors Summary: To add vectors, simply add the components separately. Use the Pythagorean theorem for the magnitude. Use trigonometry to get the angle. The vector sum will always be equal or less than the arithmetic sum of the magnitudes of the vectors.

23. 0 Physics 1710Chapter 2 Motion in One Dimension—II • The main point of today’s lecture. • A realization I had today. • A question I have. 1′ Essay:One of the following:

24. 0 Physics 1710 Chapter 3 Vectors