PHYS 172: Modern Mechanics. Summer Lecture 2 Velocity and Momentum Read:

Transcription

1 PHYS 172: Modern Mechanics Summer 2010 p sys F net t E W Q sys surr surr L sys net t Lecture 2 Velocity and Momentum Read:

2 Math Experience A) Currently taking Calculus B) Currently taking Calculus II C) Currently taking Calculus III or above D) No Calculus experience

3 Physics Experience A) Never taken a physics course B) Took an algebra-based physics course in high school C) Took a calculus-based physics course in high school D) Took an algebra-based course in college E) Took a calculus-based course in college

4 Reading Question 1 The system of units primarily used in this text is: A) feet, pounds, seconds B) kilometers, kilograms, seconds C) feet, grams, seconds D) meters, kilograms, seconds E) meters, slugs, seconds

5 Units, why bother?

6 Reading Question 2 The most general definition of momentum is: p mv A) D) p mv p mv B) E) None of the above C) p mv

7 Bonus Point Opportunity See Webassign (Must be completed by Sunday Night) The FCI assessment must be completed promptly for bonus points (both pre and post test).

8 Today: Velocity and Momentum Position Vectors Continued Vector Subtraction Relative Position Vectors Velocity Momentum

9 Identical vectors What two pieces of information do we need to create a vector? Magnitude Direction (length of the arrow) (which way the arrow points) A B A We use the symbol to denote the length (magnitude) of a vector. A

10 Scalar Multiplication When displacements and their displacement vectors have the same direction but are of different length it is very natural to think of one as being a (scalar) multiple of the other. A B 2A 2 times as far, in the same direction as A multiplication by a negative number reverses the direction of the vector. C 1.5A 1.5 times as far, in the opposite direction as A

11 Combining Displacements finish X C B C AB X start A Beginning at start, you may get to finish by Moving a displacement A, followed by moving displacement B OR by following displacement C Thus, we say C AB

12 Linear Combinations of Vectors The operations of vector addition and multiplication by scalars make it possible to express one displacement vector in terms of others. For example, this diagram indicates why we would say that C A2B. C A B Since vectors A and B have different directions they define a plane. Notice that any vector, like C, that lies in that plane can be expressed as the sum of scalar multiple of A and a scalar multiple of B. We say that C is a linear combination of A and B.

13 Linear Combinations: A 2D Example Express the vector C, shown, as a linear combination of vectors A and B, that is, find numbers a and b such that C aa bb. Solution: A C B C A B C 1.5A 0.5B components in A C basis of A, B B

14 Position Vectors Displacement vectors were introduced to describe a movement a displacement of a certain distance in a certain direction. We can use displacement vectors in a clever way: decide upon a fixed point O. Now any location can be represented by a vector that stretches from O to the point: this is called a position vector. P 2 1 r r 2 P 1 O

15 Cartesian Coordinate Systems As you know from school geometry, we can also describe the location of the point of interest by using a Cartesian coordinate system. To construct such a coordinate system we select a point, the origin of coordinates, and three mutually perpendicular axes, usually called the x, y and z axes. y Our usual coordinate system (z axis points out of page). This choice is not necessary. It is merely convenient. O x

16 Superposition & Coordinate Systems C 1A2B C A B components in basis of A, B y P 1 r xiˆ y ˆj r 1 yj ˆ 1 components in basis of î, ĵ x 1 and y 1 are also the Cartesian coordinates of point P 1. O xi 1ˆ x Close relationship between representing locations with Cartesian coordinates and with position vectors

17 CLICKER QUESTION #1 Which of these arrows represents the vector 4, 2, 0? A) B) C) D) E) a b c d e ĵ î 4, 2,0 4iˆ 2 ˆj 0kˆ

18 CLICKER QUESTION #2 A B Which of the following vectors equals? A B A B C D E None of the above

19 B A Vector Subtraction A B AB? B A A What is A B? A B A B A B B In this way we can use our tip-to-tail vector addition to perform vector subtraction.

20 Summary of 3 Vectors We ve Seen So Far 1. Displacement vectors r Represents the movement of an object from one point to another (one object at two different times) 2. Position vectors r Represents the position of an object (is defined as a vector stretching from an origin to the point of interest) 3. Relative position vectors (aka separation vectors) r 12 The vector which points from location 1 toward location 2 (two objects at the same time)

21 Velocity r r r f i Start r i y O x r f Finish

22 CLICKER QUESTION #3 A proton is at location < 0, 3, 2 > m. An electron is at location < 1, 0, 6 > m. What is the relative position vector from the proton to the electron? A) < 1, 3, 8 > m B) < 1, 3, 4 > m C) < 1, 3, 4 > m D) < 1, 3, 8 > m E) < 1, 0, 6 > m r r 1, 0, 6 m 0, 3, 2 m electron proton 10, 0 3, 6 ( 2) m 1, 3, 4 m Remember: 1, 3, 4 m is shorthand for 1 miˆ 3 m ˆj 4mk ˆ

23 Velocity r r r f i r i r f y O x Definition: average velocity v avg r rf ri t t t f i

24 Velocity 1. Speed (the magnitude of velocity, scalar) y 100 m in 10 s O x Average velocity: v avg r 100 ˆi m 10 ˆi m/s t 10 s Average speed: v avg r 100 m 10 m/s t 10 s

25 r i y 9 8m r r f Example v avg r rf ri t t t f i x 7m

26 CLICKER QUESTION #4 A bee flies in a straight line at constant speed. At 15 s after 9 AM, the bee's position is < 2, 4, 0> m. At 15.5 s after 9 AM, the bee's position is < 3, 3.5, 0> m. What is the average velocity of the bee? A) < 6, 7, 0 > m/s B) <.193,.225, 0 > m/s C) m/s D) < 0.500, , 0 > m/s E) < 2.000, , 0 > m/s v avg rf ri 3,3.5,0 m 2,4,0 m 1,.5,0 m m 2, 1,0 t 0.5s 0.5s s m 2 ˆ m 1 ˆ m i j 0 kˆ s s s

27 Object shown at equal time intervals Δt from B to E, from B to D, from B to C, v v v ave ave ave reb 3t r 2t rcb t DB As Δt 0, point C gets closer to B. When Δt 0, v ave v instantaneous

28 Instantaneous Velocity v avg v lim t 0 r t dr dt This is mathematicians being lazy

29 Given the initial position of an object + its average velocity over some time interval, we can calculate where it ends up. From the definition of average velocity we obtain the position update formula: v avg Position Update Formula r t f f r t i i rf ri vavg t f ti x x v t t f i avg, x f i y y v t t f i avg, y f i z z v t t f i avg, z f i x, y, z x, y, z v, v, v t t f f f i i i ave, x ave, y ave, z f i

30 Questions? Would you rather have to deflect a tennis ball moving at 5 meters per second (m/s) or a tennis ball moving at 100 m/s? Would you prefer to try to stop a tennis ball moving at 5 m/s or a bus moving at 5 m/s?

31 Momentum As it turns out, when it comes to interactions a more relevant quantity is momentum (we ll discuss this more next lecture). MOMENTUM for a particle of mass m and velocity v: p mv where gamma is defined as 1 1 v c 2 c m s

32 p Momentum at Small Speeds mv mv v c mv 1 when v c

33 Expressing velocity in terms of momentum we have f i avg f i r r v t t Position Update For Small Speeds at low speeds 1 t) (for small 1 2 t m p r t mc p m p r r i i f

34 Average rate of change of momentum The stronger the interaction, the faster is the change in the momentum Average rate of change of momentum: Instantaneous rate of change of momentum: p pf p t t t f dp dt i lim i t 0 p t Units: kg m 2 s

35 Today: Velocity and Momentum Position Vectors Continued Vector Subtraction Relative Position Vectors Velocity Momentum Next Time: The Momentum Principle Forces and Impulse The Momentum Principle