Chapter 3:Kinematics in 2 or 3 Dimensions; Vectors

1. Chapter 3:Kinematics in 2 or 3 Dimensions; Vectors Study Guide is posted online There will be a quiz on vectors on Friday HW3: Chap. 2: Pb. 51, Pb. 63, Pb. 67; Chap 3:Pb.3,Pb.5, Pb.10, Pb.38, Pb.46 Due Wed, Sept. 11

2. Variable Acceleration; Integral Calculus Example 2-21: Integrating a time-varying acceleration. An experimental vehicle starts from rest (v0 = 0 m/s) at t = 0s and accelerates at a rate given by a = (7.00 m/s3)t. What is (a) its velocity and (b) its displacement 2.00 s later?

3. Chapter 3:Vectors: Reading Questions • What is the difference between a vector and a scalar? • What is the magnitude of a vector? • What is the resultant of two vectors?

4. Vectors: Reading Question Which of the following statements is true: A) When you add or subtract vectors, you just add or subtract their magnitudes. B) To subtract a vector is to add its opposite. C) We never use compass headings to specify vector directions. D) The only way to add vectors is to use a ruler and a protractor.

5. r r Vector Addition: Graphical • Magnitude and Direction Magnitude only • Symbol Vectors Scalars Examples: Examples: Distance speed time Displacement Velocity acceleration

6. 2D Vectors • How do I get to Washington from New York? • Oh, it’s just 233 miles away. Magnitude and direction are both required for a vector!

7. start start Order does not matter Vector Addition: Graphical • When we add vectors B A We add vectors by drawing them “tip to tail ” The resultant starts at the beginning of the first vector and ends at the end of the second vector

8. Vector Addition Question Which graph shows the correct placement of vectors for

9. Vector Addition Question Which graph shows the correct resultant for

10. A A+ -B= -B Vector Subtraction: Graphical When you subtract vectors, you add the vector’s opposite B

11. Addition of Vectors—Graphical Methods Theparallelogram method may also be used; here again the vectors must be tail-to-tip.

12. Multiplication of a Vector by a Scalar A vectorcan be multiplied by a scalarc; the result is a vector c that has the samedirectionbut a magnitudecV.If c is negative, the resultant vector points in the opposite direction.

13. Vector Addition: Components If the components are perpendicular, they can be found usingtrigonometricfunctions.

14. A Vector Addition: Components • We don’t always carry around a ruler and a protractor, and our result isn’t always very precise even when we do. In this course we will use components to add vectors. • However, you should still always draw the vector addition to help you visualize the situation. • What are components here? Parts of the vector that lie on the coordinate axes y Ay x Ax

15. South of East C Vector Addition: Components • Once we have the components of C, Cx and Cy, we can find the magnitude and direction of C. magnitude Cx Cy direction

16. Problem 5 5. (II) V is a vector 24.8 units in magnitude and points at an angle of 23.4° above the negative x axis. (a) Sketch this vector. (b) Calculate Vx and Vy (c) Use Vx and Vy to obtain (again) the magnitude and direction of V [Note: Part (c) is a good way to check if you’ve resolved your vector correctly.]

17. Problem 6 -

18. Bx A A B B Ay Ax By C C Vector Addition: Components • We add vectors by adding their x and y components because we can add things in a line y x y By Ay Bx Ax x

19. Bx Ay Ay Ax Ax Bx By By Cx Cx Cy Cy C C Vector Addition: Components • We add vectors by adding their x and y components.

20. Unit Vectors Unit vectors have magnitude 1. Using unit vectors, any vector can be written in terms of its components:

21. Problem 16 16. (III) You are given a vector in the xy plane that has a magnitude of 90.0 units and a y component of -55units (a) What are the two possibilities for its x component? (b) Assuming the x component is known to be positive, specify the vector which, if you add it to the original one, would give a resultant vector that is 80.0 units long and points entirely in the -x direction.