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Vectors and Two-Dimensional Motion

This chapter discusses vectors and their properties, including vector addition, subtraction, and scalar multiplication. It also covers the components of vectors, unit vectors, and introduces projectile motion and relative motion.

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Vectors and Two-Dimensional Motion

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  1. Chapter 3 Vectors and Two-Dimensional Motion

  2. Vectors • Vectors – physical quantities having both magnitude and direction • Vectors are labeled either a or • Vector magnitude is labeled either |a| or a • Two (or more) vectors having the same magnitude and direction are identical

  3. R = A + B B A • Vector sum (resultant vector) • Not the same as algebraic sum • Triangle method of finding the resultant: • Draw the vectors “head-to-tail” • The resultant is drawn from the tail of A to the head of B

  4. Addition of more than two vectors • When you have many vectors, just keep repeating the process until all are included • The resultant is still drawn from the tail of the first vector to the head of the last vector

  5. Commutative law of vector addition A + B = B + A

  6. Associative law of vector addition (A + B) + C = A + (B + C)

  7. Negative vectors Vector (- b) has the same magnitude as b but opposite direction

  8. Vector subtraction Special case of vector addition: A - B = A + (- B)

  9. Multiplying a vector by a scalar • The result of the multiplication is a vector • c A = B • Vector magnitude of the product is multiplied by the scalar • |c| |A| = |B| • If the scalar is positive (negative), the direction of the result is the same as (opposite to that) of the original vector

  10. Vector components • Component of a vector is the projection of the vector on an axis • To find the projection – drop perpendicular lines to the axis from both ends of the vector – resolving the vector

  11. Vector components

  12. Unit vectors • Unit vector: • Has a magnitude of 1 (unity) • Lacks both dimension and unit • Specifies a direction • Unit vectors in a right-handed coordinate system

  13. Adding vectors by components In 2D case:

  14. Chapter 3 Problem 14 A quarterback takes the ball from the line of scrimmage, runs backwards for 10.0 yards, then runs sideways parallel to the line of scrimmage for 15.0 yards. At this point, he throws a 50.0-yard forward pass straight downfield, perpendicular to the line of scrimmage. What is the magnitude of the football’s resultant displacement?

  15. Position The position of an object is described by its position vector,

  16. Displacement The displacement of the object is defined as the change in its position,

  17. Velocity • Average velocity • Instantaneous velocity

  18. Instantaneous velocity Vector of instantaneous velocity is always tangential to the object’s path at the object’s position

  19. Acceleration • Average acceleration • Instantaneous acceleration

  20. Acceleration • Acceleration – the rate of change of velocity (vector) • The magnitude of the velocity (the speed) can change – tangential acceleration • The direction of the velocity can change – radial acceleration • Both the magnitude and the direction can change

  21. Projectile motion • A special case of 2D motion • An object moves in the presence of Earth’s gravity • We neglect the air friction and the rotation of the Earth • As a result, the object moves in a vertical plane and follows a parabolic path • The x and y directions of motion are treated independently

  22. Projectile motion – X direction • A uniform motion: ax = 0 • Initial velocity is • Displacement in the x direction is described as

  23. Projectile motion – Y direction • Motion with a constant acceleration: ay = – g • Initial velocity is • Therefore • Displacement in the y direction is described as

  24. Projectile motion: putting X and Y together

  25. Projectile motion: trajectoryand range

  26. Projectile motion: trajectoryand range

  27. Chapter 3 Problem 58 A 2.00-m-tall basketball player is standing on the floor 10.0 m from the basket. If he shoots the ball at a 40.0° angle with the horizontal, at what initial speed must he throw the basketball so that it goes through the hoop without striking the backboard? The height of the basket is 3.05 m.

  28. Relative motion • Reference frame: physical object and a coordinate system attached to it • Reference frames can move relative to each other • We can measure displacements, velocities, accelerations, etc. separately in different reference frames

  29. Relative motion • If reference frames A and B move relative to each other with a constant velocity • Then • Acceleration measured in both reference frames will be the same

  30. Questions?

  31. Answers to the even-numbered problems • Chapter 3 • Problem 2: • Approximately 484 km • (b) Approximately 18.1° N of W

  32. Answers to the even-numbered problems • Chapter 3 • Problem 6: • Approximately 6.1 units at 113° • (b) Approximately 15 units at 23°

  33. Answers to the even-numbered problems Chapter 3 Problem 10: 1.31 km north, 2.81 km east

  34. Answers to the even-numbered problems Chapter 3 Problem 28: x=7.23 × 103 m, y=1.68 × 103m

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