PHYS 100: Lecture 3. r r r VECTORS. RELATIVE MOTION in 2-D. uuur uur uur SG S W. B y j x x x C B. A y j. A x i. B x i. v W. v S.

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1 PHYS 100: Lecture 3 VECTORS A C B r r r C = A + B j i A i C B i B y j A y j C = A + B C = A + B y y y RELATIVE MOTION in 2- W S W W S SG uuur uur uur = + SG S W Physics 100 Lecture 3, Slide 1

2 Who is the Artist? A) iana Krall B) Norah Jones C) kd lang ) Madeline Peyrou E) Edith Piaf Music Haunting oice.. Highly Recommended.. Tough to categorize (at New Orleans Jazzfest, they put her in the traditional jazz tent (with Pete Fountain, etc )) BB Great ersion of Rier (joni mitchell) with kd lang Other great albums Physics 100 Lecture 3, Slide 2

3 THE BIG IEAS VECTORS: Representations Vector Addition L 2 2 L = L + L y Ly tan = L L y L L = Lcos L y = Lsin A C B r r r C = A + B What s the point? What do I epect you to know? Vectors are just math, but math that you NEE to KNOW to do physics! I epect you to MASTER this concept. It can be done with practice, practice.. RELATIVE MOTION: This topic is HAR. Why? Unfamiliar? Unnatural? NEE to LEARN to THINK in a different way!! S W SG W S W Good News? ONLY ONE EQUATION! SG = S + W uuur uur uur Physics 100 Lecture 3, Slide 3

4 WHAT I YOU FIN IFFICULT? Also, what are "i-hat, j-hat, and k-hat"? y ĵ î L r Just notation to make clear what s a ector and what s a scalar L r L y L L î = L + L y L r î ĵ L r = L î + L y ĵ L y ĵ L and L y are SCALARS L r î ĵ are VECTORS is a VECTOR of magnitude L in the +-direction is a VECTOR of magnitude L y in the +y-direction The swimming eample somewhat made sense, and I understand the relationships of elocities with respect to difference reference frames, but when it comes to actually calculating these I find that i am stumped. The equations confuse me. You are not alone.. I will try to nail the swimming problem later Physics 100 Lecture 3, Slide 4

5 Preflight 1 (A) - 60 sin(70 o ) m The displacement ector L describing the location of an object points in a direction 70 o North of West and has magnitude 60 m. Taking North to be aligned with the positie y-ais and East to be aligned with the positie -ais, What is the alue of L, the -component of L? You said: (B) (C) () (E) - 60 cos(70 o ) m + 60 cos(70 o ) m + 60 sin(70 o ) m None of the aboe y N BB Because it is west of north, the component will be negatie. We then see that there is a triangle formed with theta = 70 degrees. We then sole for the component and get the result of 56m W 70 o E L = Lcos(theta). Since theta is 70 and L is 60m, L has a magnitude of 21 m. The answer is negatie because the component points in the negatie direction. RAW A PICTURE.. THAT IS THE KEY S Physics A B 100 CLecture 3, ESlide 5

6 Preflight 1 The displacement ector L describing the location of an object points in a direction 70 o North of West and has magnitude 60 m. Taking North to be aligned with the positie y-ais and East to be aligned with the positie -ais, What is the alue of L, the -component of L? (A) (B) (C) () (E) - 60 sin(70 o ) m - 60 cos(70 o ) m + 60 cos(70 o ) m + 60 sin(70 o ) m None of the aboe L r = L î + L y ĵ y N W L y 70 o E L S L y ĵ L î L î î L negatie Physics 100 Lecture 3, Slide 6

7 Velocity & Acceleration Vectors t = 0.55 s t = 0.50 s What is the ector (t=0.45 s)? TWO ISSUES How are elocity & acceleration related?? efinitions are the KEY EFINITIONS are ALWAYS TRUE (by definition!!) d a r r r r a( t) t = ( t dt ) r r r a( t) t = ( t t) ( t 1 t 2 ) gien gien unknown How do you SUBTRACT ectors?? r r r r A B = A + ( B) A C B r r r C = A + B A -B r r r = A B Physics 100 Lecture 3, Slide 7

8 Preflight 3 You said: Since 1+2=a and the tail and head of 1 and 2 hae to be touching, the graph on b is the only one that would fit this. BB t = 0.55 s t = 0.50 s Which of these graphs represents (t=0.45 s)? I obtain (d), being my V1 ector, since I subtracted V2- a to get V1. I attached the head of 'a' to the head of V2.Therefore being able to draw the V1 ector from the tail of V2 to the tail of 'a' A B C Physics 100 Lecture 3, Slide 8

9 Preflight 3 r r r a( t) t = ( t + t) ( t t 2 gien gien unknown ) 2 t = 0.55 s t = 0.50 s Which of these graphs represents (t=0.45 s)? a r r r a( t) t + ( t + t) = ( t 1 2 -a 1 t 2 2 ) 2 1 -a Actually. a should be a t a t -a t small t large t Physics 100 Lecture 3, Slide 9

10 Relatie Motion: Reference Frames There is only one equation: TA = T A Physics 100 Lecture 3, Slide 10

11 Preflight 5 A boat is trying to cross a flowing rier. (A) At some angle upstream, against the flow You said: Which direction should the boat point in order to reach the other side of the rier in the least amount of time? (B) (C) Straight across the rier At some angle downstream, with the flow BB If he points at some angle upstream, he will offset the flow of the rier and take the most direct path across the stream. It will be the shortest distance and therefore the shortest amount of time. Since the flow of the rier is perpendicular to the shortest path across the rier it doesn't affect your ability to cross. rier flows downstream so if the boat points with the flow of water instead of against it, he will trael faster A B C Physics 100 Lecture 3, Slide 11

12 A Simpler (?) Question CALM (A) At some angle upstream A boat is trying to cross a flowing rier. Which direction should the boat point in order to reach the other side of the rier in the least amount of time? WHY I I ASK THIS QUESTION?? t (B) (C) A = t W C Straight across the rier At some angle downstream W /sin = boat A B C W = 0 t = B W boat BB BECAUSE IT S THE SAME QUESTION!! It s just posed in a different reference frame!! If W = 0, then we ARE in the reference frame of the water How does the water know if it is flowing? If it is flowing, then the houses on shore are moing past it. If it is not flowing, then the houses on shore are not moing. Physics 100 Lecture 3, Slide 12

13 The Race iewed from the Water Physics 100 Lecture 3, Slide 13

14 Relatie Motion Calculation y A swimmer, who can maintain a constant speed of S = 1.2 m/s in calm water, heads upstream at angle = 30 o. The stream measures = 50m across and the current flows at a speed of W = 0.3 m/s with respect to the shore. We want to determine how far downstream the swimmer will be when she reaches the other side. S W BB How far, in the frame of the water, does the swimmer swim to get to the other side? (A) (B) (C) () (E) /sin /cos sin cos What does the path of the swimmer look like in the water s frame? S L L cos = L L = cos Physics 100 Lecture 3, Slide 14

15 Relatie Motion Calculation y A swimmer, who can maintain a constant speed of S = 1.2 m/s in calm water, heads upstream at angle = 30 o. The stream measures = 50m across and the current flows at a speed of W = 0.3 m/s with respect to the shore. We want to determine how far downstream the swimmer will be when she reaches the other side. S L L = cos W BB How long does it take for the swimmer swim to get to the other side? (A) / S (B) (C) () /( S cos) /( S + W ) /(( S + W )cos) Calculate in water frame: Time = distance/speed Calculate in land frame: Time = y-distance/y-speed t = L = S S / cos t = = S y S cos Times are the same, as they must be! Physics 100 Lecture 3, Slide 15

16 Relatie Motion Calculation y A swimmer, who can maintain a constant speed of S = 1.2 m/s in calm water, heads upstream at angle = 30 o. The stream measures = 50m across and the current flows at a speed of W = 0.3 m/s with respect to the shore. We want to determine how far downstream the swimmer will be when she reaches the other side. t = S S cos L L = cos W BB What is the -component of the swimmer s elocity in the land frame? (A) (B) (C) () - S sin W + S sin S sin W - S sin r SG SG r = SW SW r + WG ( ) = ( ) + ( ) WG ( S ) = sin + SG S W + W Knowing this elocity and the time it takes to cross the stream, we can find how far downstream the swimmer ends up. = ( SG ) t = ( W S sin ) S cos Physics 100 Lecture 3, Slide 16