PHYS 172: Modern Mechanics Fall 2009

Transcription

1 PHYS 17: Modern Mechanics Fall 009 Lecture 17 Collisions Read

2 Clicker Question 1 Reading Question (Sections ) (This is a closed-book quiz, no consulting with neighbors, etc.) Which of the following is true of two objects that have exerienced a collision (as defined in the text): A) Total kinetic energy is conserved B) The colliding objects must come into contact C) The internal energy of the objects cannot change D) There can be no external forces E) The time of interaction must be relatively short

3 Today s Lecture Definition of collision Elastic versus inelastic collisions Head-on (1-D) collisions

4 What Qualifies as a Collision? two objects interacting during a small time interval, with little interaction before or after interval during this time interval, the interaction between the two objects is much stronger than any other external interactions Examle: System = Colliding Students (.56) Exert F = 1000 N on each other for seconds. External friction force is resent, but negligible during interaction of students.

5 What Qualifies as a Collision? Examle: Hitting a Home Run Bat and ball exert F = N on each other for 1 millisecond. The gravitational force is resent, but negligible during interaction ball and bat. Note: Both before and after collision, gravity is non-negligible. Examles: What Wouldn t Qualify as a Collision + and e in a hydrogen atom: The + -e electric interaction may be much stronger than external interactions, but Δt isn t small!

6 Energy and Momentum Conservation r r! +! = system system surroundings! E +! E = surroundings Let system = objects articiating in a collision. 0 0 always true By our definition of a collision, system and surroundings don t interact significantly during the collision. Therefore, neither momentum nor energy may be transferred between system and surroundings: r! = 0! = 0 E system system during a collision rocess (system defined as above) Now let s look at each of these conservation equations more closely.

7 Momentum Conservation in a Collision Let system = two colliding objects. 1 r r! = F! t system net, ext 0 During the collision, external forces 0. r = r f, system i, system where r = r + r system 1

8 CLICKER QUESTION # A ball bounces off a wall. v v f " i = 3kg! m / s Assume our system is the ball. What is the change in x of the ball? A) 0 kg m/s B) +3 kg m/s C) 3 kg m/s D) 6 kg m/s E) 6 kg m/s z y x Answer:! = " = " # " + # = " # ( ) ( ),, 3 kg m/s 3 kg m/s 6 kg m/s x x f x i

9 CLICKER QUESTION #3 A ball bounces off a wall. v v f " i = 3kg! m / s What is the change in x of the surroundings (the wall, Earth, etc.)? A) 0 kg m/s B) +3 kg m/s C) 3 kg m/s D) 6 kg m/s E) 6 kg m/s z y x Answer:! +! = x, sys x, surr x, surr x, sys 0! = "! = " " 6 kg # m/s = 6 kg # m/s ( )

10 Energy Conservation in a Collision Let system = two colliding objects. 0 0! E = W + Q system surr surr ( )! E =! K + K + E = 1 During the collision, external interactions 0. system 1, trans, trans internal 0 U+ K rot + K vib + E therm + E chem + We usually write this as! E =! K +! K +! E = system 1 int 0 with the understanding that ΔK 1 means ΔK 1,trans, etc.

11 Elastic and inelastic collision ELASTIC COLLISION: the internal energy of the objects in the system does not change: ΔE int =0 INELASTIC COLLISION: the internal energy of the objects in the system changes: ΔE int = 0 MAXIMALLY INELASTIC COLLISION: Objects stick together maximum energy dissiation

12 Elastic and Inelastic Collisions! E =! K +! K +! E = system If ΔE int = 0, collision is elastic. 1 int 0 The total macroscoic kinetic energy is constant. A low-energy roton collision is erfectly elastic. A billiard ball collision is aroximately elastic. If ΔE int 0, collision is inelastic. Some macroscoic kinetic energy got transformed into internal energy, or vice versa. A basketball bouncing off a wall is an inelastic collision. If two objects collide and stick together, collision is erfectly inelastic (head-on car-crash, or two lums of clay colliding).

13 A Question: energy levels for atom A An electron (K electron = 1eV) is heading straight toward atom A, which is at rest and in its ground state. Can the collision be inelastic? No. Inelastic means ΔE int 0. Only ossibility is for atom to get excited. But the necessary (threshold) energy to excite the atom is 4 ev. The incident electron isn t energetic enough to excite the atom. Further, U = 0 before and after (since atom will be far away from electron). Hence, ΔE int 0 isn t ossible here.

14 CLICKER QUESTION #4 A squishy clay ball collides in midair with a baseball, and sticks to the baseball, which kees going. Initial momenta: r r i, ball i, clay Final momentum of clay+ball: Which equation correctly describes this collision? A) B) C) r = r + r f i, ball i, clay r < r + r f i, ball i, clay r > r + r f i, ball i, clay r f Answer: In a collision, P sys is always conserved (only if system = colliding objects)

15 Elastic Head-On (1-D) Collisions Initial (just before collision) 1 M r r = 0 Final (just after collision) Momentum conservation: r r r! P =! +! = tot m M r r r r ( ) ( ) 0 3! 1 + 4! = = + 1x 3x 4x two unknowns r 3 r 4 Energy conservation:! E =! K +! K +! E = 0 sys m M int ( K K ) ( K K ) elastic 3! 1 + 4! = 0! 3x 1 x "! " 4x $ # % + $ # 0 % = 0 & m m ' & M ' same two unknowns

16 Elastic Head-On (1-D) Collisions r M r = 0 1 r 3 r 4 = + 1x 3x 4x 1 x 3x 4x = + m m M unknowns, equations: we can solve for 3x and 4x. Note: if collision had been inelastic, we d have too many unknowns! Do algebra. For m M, result is a quadratic equation with two solutions: Solution #1: = and = 0 3x 1x 4x (no collision! objects not lined u) Solution #: " m! M # = $ % and v = m v & M + m ' M + m 3x 1x 4x 1x m M here Note: 3x < 1x (as it must be it gave some momentum to the other ball)

17 Two Secial Cases (remember m M here) M r r = 0 1 r 3 r 4 " m! M # = $ % and v = m v & M + m ' M + m 3x 1x 4x 1x Case I: m M! 0 and v = v 3x 4x 1x First ball stos in lace, second ball moves ahead with same seed. Case II: m << M! " and v! 0 3x 1x 4x First ball stos bounces straight back, second ball doesn t move.

18 A Concluding Puzzle James Clerk Maxwell once walked into a ool hall to observe a game of billiards. He remarked that he knew with certainty, without making any measurements, that the collisions between the billiard balls were inelastic. How did he know this?

19 EXTRA CLICKER QUESTIONS

20 CLICKER QUESTION #5 A squishy clay ball collides in midair with a baseball, and sticks to the baseball, which kees going. Initial kinetic energies: K i,clay K i,ball Which equation correctly describes this collision? A) K f = K i,clay + K i,ball B) K f < K i,clay + K i,ball Final kinetic energy of clay+ball: C) K f > K i,clay + K i,ball K f Answer: In an inelastic collision, some K is converted to E int (or vice versa).

21 A bullet of mass m traveling horizontally at a very high seed v embeds itself in a block of mass M that is sitting at rest on a nearly frictionless surface. What is the seed of the block just after the bullet embeds itself in the block? m M + A) v B) C) m D) E) v Answer: CLICKER QUESTION #6 m M + M + m m ( ) v m r r r! = 0 " + = v m M sys i, bullet i, block f, bullet+ block mv + M # 0 = m + M v v f = m m + M v f v

22 CLICKER QUESTION #7 Which of the following equations is the roer energy balance for this collision, for the system of bullet + block? 1 1 m )v f mv i A) ( M + = B) 1 1 C) Mv = D) f mv i 1 (M + m)v f +!E internal = 1 mv i!e internal = 1 mv i E) 1 (M + m)v f = 1 mv i +!E internal ANSWER: K + E = K + K + E f, bullet+ block f,int i, bullet i, block i,int K +! E = K f, bullet+ block int i, bullet

23 CLICKER QUESTION # After the collision the ing-ong ball's kinetic energy is K. What is the kinetic energy of the bowling ball? A ing-ong ball bounces elastically off a bowling ball which is initially at rest. A) K B) -K C) much greater than K D) negligibly small (nearly zero)

24 CLICKER QUESTION # A ball of mass m 1 hits a stationary target of mass m head-on. The total initial and final kinetic energies are the same. Which of the following statements is false? A) If m 1 << m, the momentum of the ball hardly changes. B) If m 1 < m, the ball bounces straight back. C) If m 1 < m, the ball bounces straight back with less kinetic energy than it had originally. D) If m 1 >> m, the ball kees going without change of direction.

25 CLICKER QUESTION # Two lead bricks moving in the +x and x directions, each with kinetic energy K, smash into each other and come to a sto. What haened to the energy? A) The kinetic energy of the system remained constant. B) The kinetic energy changed into thermal energy. C) The total energy of the system decreased by an amount K. D) Since the blocks were moving in oosite directions, the initial kinetic energy of the system was zero, so there was no change in energy.