PHYS 1443 – Section 002 Lecture #5 Wednesday, Sept. 12, 2007 Dr. Jaehoon Yu • • • Coordinate Systems Vectors and Scalars Motion in Two Dimensions – Motion under constant acceleration – Projectile Motion – Maximum ranges and heights Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 1 Announcements • E-mail distribution list: 63 of you subscribed to the list so far – Please subscribe to the class distribution list since this is the primary communication tool for this class • Two physics department seminars this week – At 4pm today in SH101 • Dr. Sandy Dasgupta, UTA Chemistry Department chair: Playing With Drops and Films Analytical Chemistry in Small Places – At 4pm Friday in SH101 • Dr. Amitava Patra: Nano Materials for Photonic Application Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 2 Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 3 Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 4 Special Problems for Extra Credit • Derive the quadratic equation for Bx2-Cx+A=0 5 points • Derive the kinematic equation v vxi 2axx x from first principles and the known kinematic equations 10 points • You must show your work in detail to obtain full credit • Due Wednesday, Sept. 19 2 xf Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 2 f i 5 1D Kinematic Equations of Motion on a Straight Line Under Constant Acceleration vxf t vxi axt Velocity as a function of time 1 1 xf xi v x t vxf vxi t Displacement as a function of velocities and time 2 2 1 2 xf xi vxit axt 2 vxf vxi 2axxf xi 2 2 Displacement as a function of time, velocity, and acceleration Velocity as a function of Displacement and acceleration You may use different forms of Kinematic equations, depending on the information given to you for specific physical problems!! Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 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, q (r,q) • Vectors become a lot easier to express and compute +y How are Cartesian and Polar coordinates related? y1 (x1,y1) =(r1,q1) r1 x1 r1 cos q1 r x12 y12 1 q1 O (0,0) Wednesday, Sept. 12, 2007 x1 +x y1 r1 sin q1 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu y1 tan q1 x1 y1 x 1 q1 tan 1 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. r y q qs (-3.50,-2.50)m 2 y2 3.50 2 2.50 2 18.5 4.30( m) x r x q 180 q s tan q s 2.50 5 3.50 7 1 5 q s tan 35.5 7 o o o q 180 q s 180 35.5 216 Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu o 8 Vector and Scalar Vector quantities have both magnitudes (sizes) and directions Force, gravitational acceleration, momentum ur Normally denoted in BOLD letters, F, or a letter with arrow on top F Their sizes or magnitudes are denoted with normal letters, F, or ur absolute values: F or F Scalar quantities have magnitudes only Can be completely specified with a value and its unit Normally denoted in normal letters, E Energy, heat, mass, time Both have units!!! Wednesday, Sept. 12, 2007 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 9 Properties of Vectors • Two vectors are the same if their sizes and the directions are the same, no matter where they are on a coordinate system!! Which ones are the same vectors? y D A=B=E=D F A Why aren’t the others? B x E Wednesday, Sept. 12, 2007 C PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu C: The same magnitude but opposite direction: C=-A:A negative vector F: The same direction but different magnitude 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+C+D+E=E+C+A+B+D A+B B A • A = B A+B OR A+B B A Subtraction: – The same as adding a negative vector:A - B = A + (-B) A A-B • -B Since subtraction is the equivalent to adding a negative vector, subtraction is also commutative!!! Multiplication by a scalar is increasing the magnitude A, B=2A Wednesday, Sept. 12, 2007 B 2A A PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu B=2A 11 Example for Vector Addition A car travels 20.0km due north followed by 35.0km in a direction 60.0o west of north. Find the magnitude and direction of resultant displacement. r Bsinq N B 60o Bcosq r q 20 A 2 B sin q 2 A2 B 2 cos 2 q sin 2 q 2 AB cos q A2 B 2 2 AB cosq 20.02 35.02 2 20.0 35.0 cos 60 2325 48.2(km) E B sin 60 q tan 1 tan 1 35.0 sin 60 20.0 35.0 cos 60 30.3 38.9 to W wrt N 37.5 tan 1 Wednesday, Sept. 12, 2007 A B cosq A B cos 60 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu Find other ways to solve this problem… 12 Components and Unit Vectors Coordinate systems are useful in expressing vectors in their components y (+,+) Ay A (-,+) (Ax,Ay) q (-,-) u r A (+,-) Ax ur A cos q 2 Ax ur A cos q Ay ur A sin q x 2 ur A sin q u r 2 A cos 2 q sin 2 q Wednesday, Sept. 12, 2007 A Ax Ay PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 2 } Components } Magnitude 2 u r A 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 r r r i, j, k So the vector A can be re-written as Wednesday, Sept. 12, 2007 r ur ur r ur r r A Ax i Ay j A cos q i A sin q j PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 14 Examples of Vector Operations Find the resultant vector which is the sum of A=(2.0i+2.0j) and B =(2.0i-4.0j) r r r r ur ur ur C A B 2.0i 2.0 j 2.0i 4.0 j r 2.0 2.0 i ur C 4.0 2.0 2 r 2.0 4.0 j r r 4.0i 2.0 j m 2 16 4.0 20 4.5(m) q 2.0 tan tan 27o Cx 4.0 1 Cy 1 Find the resultant displacement of three consecutive displacements: d1=(15i+30j +12k)cm, d2=(23i+14j -5.0k)cm, and d3=(-13i+15j)cm ur ur ur ur r r r r r r r r D d 1 d 2 d 3 15i 30 j 12k 23i 14 j 5.0k 13i 15 j r r r r r r 15 23 13 i 30 14 15 j 12 5.0 k 25i 59 j 7.0k (cm) Magnitude Wednesday, Sept. 12, 2007 ur D 25 59 7.0 65(cm) 2 2 PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 2 15 Displacement, Velocity, and Acceleration in 2-dim • Displacement: • Average Velocity: • Instantaneous Velocity: • Average Acceleration • Instantaneous Acceleration: Wednesday, Sept. 12, 2007 r r r r r f r i r r r r r f ri r v t f ti t r r r r dr v lim t 0 t dt r r r r v f vi v a t f ti t How is each of these quantities defined in 1-D? r r r r 2 r v dv d d r d r a lim 2 dt dt t 0 t dt dt PHYS 1443-002, Fall 2007 Dr. Jaehoon Yu 16 Kinematic Quantities in 1D and 2D Quantities 1 Dimension 2 Dimension r r x x f xi Displacement Average Velocity r r r r r r f r i v t t f ti x f xi x vx t t f ti Inst. Velocity x dx v x lim t 0 t dt Average Acc. v x v xf v xi ax t t f ti Inst. Acc. What is the Wednesday, Sept. 12, 2007 r r r f ri r r r r d r v lim t 0 t dt r r r r v v f vi a t t f ti r r r 2 vx dvx d x r v dv d r ax lim 2 a lim 2 t 0 t t 0 t dt dt dt dt 2 difference between 1D2007 and 2D quantities? PHYS 1443-002, Fall Dr. Jaehoon Yu 17