Physics 1710 Chapter 3 Vectors Demonstration Egg Toss

Physics 1710 Chapter 3 Vectors Demonstration: Egg Toss REVIEW

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

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

Physics 1710 Chapter 3 Vectors Seat belt: Air Bag Video

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.

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

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 The driver between replies, “Oh! Now it is very to Budapest. ” distancefar and displacement.

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

Physics 1710 Chapter 3 Vectors A Scalar is 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.

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.

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

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.

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

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∣ = √ [Ax 2 + Ay 2 + Az 2 ]

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

Physics 1710 Chapter 2 Motion in One Dimension—II 2 -Dimensional Vector Y(m) x Position Vector r | = r = √(x 2 + y 2) r y = √(2. 0 2 + 1. 5 2) = √(4. 0 + 2. 25 ) = √(6. 25) = 2. 5 m X (m)

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

Physics 1710 Chapter 3 Vectors 80/20 Fact: 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 A x Cy = k A y Cz = k A z

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

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 = A 2 +2 AB +B 2 LHS = Ax 2 +Ay 2 + Bx 2 + By 2 + 2 Ax. Bx + 2 Ay. By RHS = Ax 2 + Ay 2+ Bx 2 + By 2 +2√(Ax 2 Bx 2 + Ay 2 By 2 + Ay 2 Bx 2 + Ax 2 By 2) LHS ≤ RHS 2 Ax. Bx + 2 Ay. By≤ 2√(Ax 2 Bx 2 + Ay 2 By 2 + Ay 2 Bx 2 + Ax 2 By 2) A x 2 B x 2 + 2 A x B x A y B y + A y 2 B y 2 ≤ A x 2 B x 2 + A y 2 B y 2 + A y 2 B x 2 + A x 2 B y 2 2 A x B x A y. B y ≤ A y 2 B x 2 + A x 2 B y 2 iff 0 ≤ (Ay. Bx - Ax By) 2

Physics 1710 Chapter 3 Vectors 80/20 Fact: 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)

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.

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

Physics 1710 Chapter 3 Vectors