Chapter Three

1. Chapter Three Vectors and Two-Dimensional Motion

2. Scalars have only size. Vectors have both size and direction Examples of scalars: Examples of vectors • Equality of Two Vectors • Two vectors are equal if they have the same magnitude and the same direction • Movement of vectors in a diagram • Any vector can be moved parallel to itself without being affected Section 3.1

3. Distance vs Displacement 5 meters east in 2 seconds 3 meters south in 1 second The total distanced traveled is The total displacement d is Speed = d / t = v Velocity= d / t = v

4. Vector mathematics is different than scalar mathematics Scalars are just numbers and can be added with regular math. They add up to a sum. Vectors have direction and must be drawn to scale and connected head-to-tail. They add up to a resultant. HEAD Tail Start Finish

5. 2 miles 3 miles 4 miles Hawkeye travels 5 miles as shown before Natasha Romanov sends him back. He takes a shortcut back that saves him a mile. If the whole round trip took 1 hour, what is his a) speed? b) displacement ?

6. Practice; 1. A runner runs 500 meters due east and 1000 meters due south. Find the distance and displacement 2. A plane travels 3000 miles from California to New York in 5 hours, stays in NY 1 hour to refuel, then flies back again. Find the average speed and average velocity for the round trip flight .

7. When direction changes, average velocity = average speed Ex; Wilbert drives 2.0 km east in 15 minutes to make a delivery. He then goes 3.0 km west in 20 minutes to make a delivery to the other side of town. Find Wilberts’ a) average speed b) average velocity During his 35 minutes of travel

8. Graphically Adding Vectors • draw the vectors, connecting them “head-to-tail” • The resultant is drawn from the origin of the first vector to the end of the last vector • Measure the length of the resultant and its angle • Use the scale factor to convert length to actual magnitude Section 3.1

9. Notes about Vector Addition • Vectors obey the Commutative Law of Addition • The order in which the vectors are added doesn’t affect the result Section 3.1

10. Graphically Adding Vectors, cont. • When you have many vectors, just keep repeating the “tip-to-tail” process until all are included • The resultant is still drawn from the origin of the first vector to the end of the last vector Section 3.1

11. Relative Velocity, Example • Need velocities • Boat relative to river • River relative to the Earth • Boat with respect to the Earth (observer) • Equation

12. 100 m  V boat = 2 m/s Vriver = 5 m/s A boat tries to cross a 100 m wide river. The boat’s engine can move it at 2 m/s and the river flows at 5 m/s. Find the resultant velocity, trajectory displacement a of the boat and time to cross the river.

13. Components of a Vector • It is useful to use rectangular components to add vectors • These are the projections of the vector along the x- and y-axes Section 3.2

14. Components of a Vector, cont. • The x-component of a vector is the projection along the x-axis • The y-component of a vector is the projection along the y-axis • Then, Section 3.2

15. Projectile Motion Section 3.4

16. Some Variations of Projectile Motion • An object may be fired horizontally • The initial velocity is all in the x-direction • vo = vx and vy = 0 • All the general rules of projectile motion apply Section 3.4

17. Projectile problem 1 A projectile is fired straight off of a 100 m high cliff at 50 m/s. Find • the time it takes to hit the ground b) the horizontal distance it will travel c) The velocity when it hits the ground. • Use d y = viyt + ½ ayt2 • Use Vx = dx/t • Use for Vyf: Vf = Vi + at Then add head to tail with Pythagorean theorem to find the total V final

18. Projectile problem 2 A ball is hit at a 36º angle upward and leaves the bat at 25 m/s. Find • the initial horizontal speed b) the initial vertical speed c) the time it takes to hit the ground • the height it will travel e) the horizontal distance it will travel • Use Vx = 25 Cos36 b) Use Vy = 25 sin 36 c) Use Vfy = 0 = Viy +at d) Use dy = 0 + ½ at2 e) Use dx = Vxt

19. Hints for 2D projectile problems • Separate given data into x and y. • Finding time in the air is a good first step. Find an equation with t and the things you are given. • ay is always 9.8 m/s2. • Sometimes Vx =0 or Vy = 0. • Vy always equals zero at the top of a parabola.

20. Projectile Motion • An object may move in both the x and y directions simultaneously • It moves in two dimensions • The form of two dimensional motion we will deal with is an important special case called projectile motion Section 3.4

21. Assumptions of Projectile Motion • We may ignore air friction • We may ignore the rotation of the earth • With these assumptions, an object in projectile motion will follow a parabolic path Section 3.4

22. Rules of Projectile Motion • The x- and y-directions of motion are completely independent of each other • The x-direction is uniform motion • ax = 0 • The y-direction is free fall • ay = -g • The initial velocity can be broken down into its x- and y-components Section 3.4

23. Projectile Motion at Various Initial Angles • Complementary values of the initial angle result in the same range • The heights will be different • The maximum range occurs at a projection angle of 45o Section 3.4

24. Non-Symmetrical Projectile Motion • Follow the general rules for projectile motion • Break the y-direction into parts • up and down • symmetrical back to initial height and then the rest of the height Section 3.4

25. Special Equations • The motion equations can be combined algebraically and solved for the range and maximum height Section 3.4

26. Relative Velocity • Relative velocity is about relating the measurements of two different observers • It may be useful to use a moving frame of reference instead of a stationary one • It is important to specify the frame of reference, since the motion may be different in different frames of reference • There are no specific equations to learn to solve relative velocity problems Section 3.5

27. Relative Velocity Notation • The pattern of subscripts can be useful in solving relative velocity problems • Assume the following notation: • E is an observer, stationary with respect to the earth • A and B are two moving cars Section 3.5

28. Relative Position Equations • is the position of car A as measured by E • is the position of car B as measured by E • is the position of car A as measured by car B Section 3.5

29. Relative Position • The position of car A relative to car B is given by the vector subtraction equation Section 3.5

30. Relative Velocity Equations • The rate of change of the displacements gives the relationship for the velocities Section 3.5

31. Problem-Solving Strategy: Relative Velocity • Label all the objects with a descriptive letter • Look for phrases such as “velocity of A relative to B” • Write the velocity variables with appropriate notation • If there is something not explicitly noted as being relative to something else, it is probably relative to the earth Section 3.5

32. Problem-Solving Strategy: Relative Velocity, cont • Take the velocities and put them into an equation • Keep the subscripts in an order analogous to the standard equation • Solve for the unknown(s) Section 3.5