1. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 1 PHYS 1441 – Section 002 Lecture #5 Wednesday, Feb. 11, 2009 Dr. Jaehoon Yu Coordinate systems Vectors and Vector Operations Motion in Two Dimensions Motion under constant acceleration Projectile Motion

2. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 2 Announcements E-mail distribution list: 58 of you subscribed to the list –Please be sure to subscribe to the list as soon as possible. First term exam –1 – 2:20pm, Wednesday, Feb. 18 –Covers: CH1.1 – what we complete on Monday, Feb. 16 + appendix A1 – A8 Physics Department colloquium scheduled at 4pm today in SH101 –There is a double extra credit for colloquium today

3. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 3

4. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 4 Reminder: Special Problems for Extra Credit Derive the quadratic equation for yx 2 -zx+v=0  5 points Derive the kinematic equation from first principles and the known kinematic equations  10 points You must show your work in detail to obtain the full credit Due next Monday, Feb. 16

5. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 5 Kinematic Equations of Motion on a Straight Line Under Constant Acceleration Velocity as a function of time Displacement as a function of velocities and time Displacement as a function of time, velocity, and acceleration Velocity as a function of Displacement and acceleration You may use different forms of Kinetic equations, depending on the information given to you for specific physical problems!!

6. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 6 Coordinate Systems They make it easy and consistent to express locations or positions Two commonly used systems, depending on convenience, are –Cartesian (Rectangular) Coordinate System Coordinates are expressed in (x,y) –Polar Coordinate System Coordinates are expressed in distance from the origin ® and the angle measured from the x-axis,  (r  ) Vectors become a lot easier to express and compute O (0,0) (x 1,y 1 ) r1r1  How are Cartesian and Polar coordinates related? y1y1 x1x1 +x +y = (r 1   )

7. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 7 Example Cartesian Coordinate of a point in the xy plane are (x,y)= (-3.50,-2.50)m. Find the equivalent polar coordinates of this point. y x (-3.50,-2.50)m r  ss

8. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 8 Vector and Scalar Vector quantities have both magnitudes (sizes) and directions Scalar quantities have magnitudes only Can be completely specified with a value and its unit Force, gravitational acceleration, momentum Normally denoted in BOLD letters, F F, or a letter with arrow on top Their sizes or magnitudes are denoted with normal letters, F, or absolute values: Energy, heat, mass, time Normally denoted in normal letters, E Both have units!!!

9. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 9 Properties of Vectors Two vectors are the same if their and the are the same, no matter where they are on a coordinate system!! x y A B E D C F Which ones are the same vectors? A=B=E=D Why aren’t the others? C: C: The same magnitude but opposite direction: C=-A: C=-A: A negative vector F: F: The same direction but different magnitude sizes directions

10. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 10 Vector Operations Addition: –Triangular Method: One can add vectors by connecting the head of one vector to the tail of the other (head-to-tail) –Parallelogram method: Connect the tails of the two vectors and extend –Addition is commutative: Changing order of operation does not affect the results A+B=B+A A+B=B+A, A+B+C+D+E=E+C+A+B+D A B A B = A B A+B Subtraction: –The same as adding a negative A vector: A - B = A + B (- B )A -B Since subtraction is the equivalent to adding a negative vector, subtraction is also commutative!!! Multiplication by a scalar is increasing the magnitude A, BA B =2 AA B=2A A+B A+B A-B OR

11. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 11 Example for Vector Addition A car travels 20.0km due north followed by 35.0km in a direction 60.0 o west of north. Find the magnitude and direction of resultant displacement. N E    r 20A B Find other ways to solve this problem… Bcos  Bsin 

12. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 12 Components and Unit Vectors Coordinate systems are useful in expressing vectors in their components (A x,A y ) A  AyAy AxAx x y } Components (+,+) (-,+) (-,-)(+,-) } Magnitude

13. Wednesday, Feb. 11, 2009PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 13 Unit Vectors Unit vectors are the ones that tells us the directions of the components Dimensionless Magnitudes are exactly 1 Unit vectors are usually expressed in i, j, k or So the vector A can be re-written as

14. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 14 Examples of Vector Operations Find the resultant vector which is the sum of Aij A =(2.0 i +2.0 j ) and B ij =(2.0 i -4.0 j ) Find the resultant displacement of three consecutive displacements: d 1 ij d 1 =(15 i +30 j k +12 k )cm, d 2 ij d 2 =(23 i +14 j k -5.0 k )cm, and d 3 ij d 3 =(-13 i +15 j )cm Magnitude

15. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 15 2 Dimensional Kinematic Quantities

16. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 16 Average velocity is the displacement divided by the elapsed time. 2D Average Velocity

17. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 17 The instantaneous velocity indicates how fast the car moves and the direction of motion at each instant of time.

18. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 18 2D Instantaneous Velocity

19. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 19 2D Average Acceleration

20. Wednesday, Feb. 11, 2009 PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 20 Displacement, Velocity, and Acceleration in 2-dim Displacement: Average Velocity: Instantaneous Velocity: Average Acceleration Instantaneous Acceleration: How is each of these quantities defined in 1-D?

21. Wednesday, Feb. 11, 2009PHYS 1441-002, Spring 2009 Dr. Jaehoon Yu 21 Kinematic Quantities in 1D and 2D Quantities1 Dimension2 Dimension Displacement Average Velocity Inst. Velocity Average Acc. Inst. Acc. What is the difference between 1D and 2D quantities?

22. Wednesday, Feb. 6, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 22 This is a motion that could be viewed as two motions combined into one. A Motion in 2 Dimension

23. Wednesday, Feb. 6, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 23 Motion in horizontal direction (x)

24. Wednesday, Feb. 6, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 24 Motion in vertical direction (y)

25. Wednesday, Feb. 6, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 25 Imagine you add the two 1 dimensional motions on the left. It would make up a one 2 dimensional motion on the right. A Motion in 2 Dimension

26. Wednesday, Feb. 6, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 26 x-component Kinematic Equations in 2-Dim y-component

27. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 27 In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s 2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s 2. Find (a) x and vx,vx, (b) y and v y, and (c) the final velocity of the spacecraft at time 7.0 s. Ex. A Moving Spacecraft

28. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 28 How do we solve this problem? 1.Visualize the problem  Make a drawing. 2.Decide which directions are to be called positive (+) and negative (-). 3.Write down the values that are given for any of the five kinematic variables associated with each direction. 4.Verify that the information contains values for at least three of the kinematic variables. Do this for x and y separately. Select the appropriate equation. 5.When the motion is divided into segments, remember that the final velocity of one segment is the initial velocity for the next. 6.Keep in mind that there may be two possible answers to a kinematics problem.

29. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 29 In the x direction, the spacecraft has an initial velocity component of +22 m/s and an acceleration of +24 m/s 2. In the y direction, the analogous quantities are +14 m/s and an acceleration of +12 m/s 2. Find (a) x and vx,vx, (b) y and v y, and (c) the final velocity of the spacecraft at time 7.0 s. xaxax vxvx v ox t ?? yayay vyvy v oy t ?? Ex. continued +24.0 m/s 2 +22.0 m/s +14.0 m/s+12.0 m/s 2 7.0 s

30. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 30 xaxax vxvx v ox t ?+24.0 m/s 2 ?+22 m/s7.0 s First, the motion in x-direciton…

31. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 31 yayay vyvy v oy t ?+12.0 m/s 2 ?+14 m/s7.0 s Now, the motion in y-direction…

32. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 32 The final velocity… A vector can be fully described when the magnitude and the direction are given. Any other way to describe it? Yes, you are right! Using components and unit vectors!!

33. Monday, Feb. 11, 2008PHYS 1441-002, Spring 2008 Dr. Jaehoon Yu 33 If we visualize the motion…