1. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 1 PHYS 1443 – Section 501 Lecture #4 Monday, Feb. 2, 2004 Dr. Andrew Brandt Motion in Two Dimensions Vector Properties and Operations Motion under constant acceleration

2. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 2 Announcements Homework: 38/44 of you have signed up (11 have submitted some answers) Keep enough significant figures, get sign right HW1 dues Weds at midnight (questions before class?)

3. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 3 Kinetic Equations of Motion in a Straight Line Under Constant Acceleration Velocity as a function of time Displacement as a function of velocity 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 problems!!

4. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 4 Solving Problems (p.28) Read problem carefully Draw a diagram (choose coordinate system) Write down known and unknown quantities Think! Calculate (substitute numbers at end) Is answer reasonable? Check units Air Jordan revisited

5. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 5 Coordinate Systems (Start Ch. 3) Makes it easy to express locations or positions Two commonly used systems, depending on convenience –Cartesian (Rectangular) Coordinate System Coordinates are expressed in (x,y) –Polar Coordinate System Coordinates are expressed in (r  ) Vectors become a lot easier to express and compute O (0,0) (x 1,y 1 )=(r  ) r  How are Cartesian and Polar coordinates related? y1y1 x1x1 +x +y

6. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 6 Example Cartesian Coordinate of a point in the xy plane are (x,y)= (-3.50,- 2.50)m. Find the polar coordinates of this point. y x (-3.50,-2.50)m r  ss

7. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 7 Vector and Scalar Vector quantities have both magnitude (size) and direction Scalar quantities have magnitude only Can be completely specified with a value and its unit Force, acceleration, momentum BOLD F Normally denoted in BOLD letters, F, or a letter with arrow on top Their sizes or magnitudes are denoted with normal letters, F, or absolute values: Energy, heat, mass, speed Normally denoted in normal letters, E Both have units!!!

8. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 8 Properties of Vectors Two vectors are the same if their and 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=-A: C: The same magnitude but opposite direction: C=-A: A negative vector F: F: The same direction but different magnitude sizesdirections

9. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 9 Vector Operations Addition: –Triangular Method: One can add vectors by connecting the tail of one vector to the tip of the other (tail-to-tip) –Parallelogram method: Connect the tails of the two vectors and extend A+B=B+AA+B+C+D+E=E+C+A+B+D –Addition is commutative: Changing order of operation does not affect the results A+B=B+A, A+B+C+D+E=E+C+A+B+D A B A B = A B A+B Subtraction: A B = A B –The same as adding a negative vector: A - B = A + (- B )A -B A, BAMultiplication by a scalar is increasing the magnitude A, B =2 AA B=2A A+B A+B A-B OR

10. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 10 Vector Components Coordinate systems are useful for resolving vectors into their components (lengths along coordinate axes…) This makes vector algebra easier Don’t forget about direction of vectors (A x,A y ) A  AyAy AxAx x y } Components (+,+) (-,+) (-,-)(+,-) } Magnitude Example on board

11. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 11 Unit Vectors Unit vectors indicate the directions of the components Dimensionless Magnitudes are exactly 1 Unit vectors are usually expressed as i, j, k for x, y, and z axis, respectively or A So the vector A can be re-written as

12. Monday, February 2, 2004PHYS 1443-501, Spring 2004 Dr. Andrew Brandt 12 Examples of Vector Addition AijB ij Find the resultant vector which is the sum of A =(2.0 i +2.0 j ) and B =(2.0 i -4.0 j ) d 1 ij kd 2 ij kd 1 ij Find the resultant displacement of three consecutive displacements: d 1 =(15 i +30 j +12 k )cm, d 2 =(23 i +14 j -5.0 k )cm, and d 1 =(-13 i +15 j )cm OR Magnitude Direction Magnitude