AP Physics C Mechanics Vectors

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2 AP Physics C Mechanics Vectors

3 Scalar Versus Vector A scalar has only a physical quantity such as mass, speed, and time. A vector has both a magnitude and a direction associated with it, such as velocity and acceleration. A vector is denoted by an arrow above the variable, 3

4 1 Is this a vector or a scalar? Time Speed Velocity Distance Displacement Scalar Scalar Vector Scalar Vector 4

5 2 Which of the following is a true statement? A B C D E It is possible to add a scalar quantity to a vector. The magnitude of a vector can be zero even though one of its components is not zero. The sum of the magnitude of two unequal vectors can be zero. Rotating a vector about an axis passing through the tip of the vector does not change the vector. Vectors must be added geometrically. 5

6 Drawing A Vector A vector is always drawn with an arrow at the tip indicating the direction, and the length of the line determines the magnitude. Remember displacement is the distance away from your initial position, it does not account for the actual distance you moved. 6

7 Determining Magnitude and Direction anti parallel All of these vectors have the same magnitude, but vector B runs anti parallel therefore it is denoted negative A. 7

8 Vector Addition 8

9 Vector Addition Methods Tail to Tip Method 9

10 Vector Addition Methods Parallelogram Method Place the tails of each vector against one another. Finish drawing the parallelogram with dashed lines and draw a diagonal line from the tails to the other end of the parallelogram to find the vector sum. 10

11 3 If a car under goes a displacement of 3 km North and another of 4 km to the East what is the net displacement? A 5 2 km 4 km B C 5 km 4 3 km 3 km x D 7 km E 6 km 11

12 4 If a car under goes a displacement of 3 km North and another of 4 km to the East what is the total distance traveled? A 5 2 km 4 km B C 7 km 5 km 3 km x D 4 km E 3 km 12

13 5 Solve for θ A B C D E 45 o 75 o 53 o 37 o 25 o 3 km θ 4 km x 13

14 Vector Components v y v θ v x A vector that makes an angle with the axis has both a horizontal and vertical component of velocity. θ is measured starting at the x axis and rotating in the direction of the y axis. 14

15 Multiple Vectors When dealing with multiple vectors you can just add the components in order to attain the components of the vector sum. v x v x v y v y v x v y v y v x 15

16 6 The components of vector A are given as follows: The magnitude of A is closest to: A 4.2 B 8.4 C 11.8 D 18.9 E

17 7 The components of vectors are given as follows: and Solve for the magnitude of A 5 B 17 C 17 D 10 E 8 17

18 8 The components of vector A are given as follows: The angle measured counter clockwise from the x axis to vector A, in degrees, is closest to: A B C D E 339 o 200 o 122 o 21 o 159 o 18

19 9 The components of vector A and B are given as follows: The magnitude of B A, is closest to: A B 4.92 C 2.8 D 9.7 E 25 19

20 10 The magnitude of B is 5.2. Vector B lies in the 4th quadrant and forms a 30 o with the x axis. The components of B x and B y are: A B C D 20

21 11 The magnitude of vector A is equal to vector B plus vector C. What is the value of vector A? A 2.59 B 1.78 C 3.42 D 1.63 E O y 45 O x 6 21

22 12 Vectors A and B are shown. Vector C is given by C = A + B. In the figure above, the magnitude of C is closest to: A 7.5 B 3.9 C 5.2 D 9.3 E o 60 o 22

23 Unit Vectors Unit vectors have no units and a magnitude of 1. Unit Vectors describe a direction in space. indicates the x direction indicates the y direction indicates the z direction Any given Vector can be presented in terms of unit vectors: 23

24 Unit Vectors When two vectors A and B are presented in terms of their components, we can express the vector sum R using unit vectors: 24

25 13 What is the magnitude of the sum of the following vectors? A 9.3 B 12.3 C 5.1 D 10.7 E 3 25

26 Products of Vectors Scalar Product also known as Dot Product yields a scalar quantity value can be positive, zero, or negative depending on θ. θ ranges from 0 to 180 degrees. = = = = = = 26

27 14 In the figure, find the scalar product of vectors B and C, A 0 B 17 C 24 D 17 E o 4 45 o 6 27

28 15 In the figure, find the scalar product of vectors A and C, A 0 B 14 C 42 D 14 E o 4 45 o 6 28

29 Products of Vectors Vector Product also known as the cross product yields another vector. = = = = = = = = = 29

30 16 In the figure, find the vector product of vectors A and B. A 12 B 30 C 25 D 20 E o 4 45 o 6 30

31 17 Two vectors are give as follows: Solve for A B C D E 31

32 18 Two vectors are give as follows: Solve for A 2 B 4 C 7 D 5 E 12 32

33 19 Which of the following is an accurate statement? A B If the vectors A and B are each rotated through the same angle about the same axis, the product will be unchanged. If the vectors A and B are each rotated through the same angle about the same axis, the product A x B will be unchanged C D If a vector A is rotated about an axis parallel to vector B, the product will be changed. When a scalar quantity is added to a vector, the result is a vector of larger magnitude than the original vector. 33

34 20 Solve for the angle between vector and A B C D E o o 57 o o o 34

35 21 Two vectors are given: The angle between vectors A and B, in degrees, is: A B C D E 117 o 76 o 150 o 29 o 161 o 35

36 22 Two vectors are given: Solve for the magnitude of A 33 B 29 C 25 D 21 E 17 36