Conceptual Physics Energy Sources Collisions
Conceptual Physics Energy Sources Collisions
Lana Sheridan
De Anza College
July 13, 2017 Last time
• energy and work
• potential energy
• conservation of energy
• energy transfer
• simple machines
• efficiency Overview
• discussion of energy sources
• momentum and collisions Sources of Energy
Discussion! Sources of Energy
Sources of energy: • oil • coal • natural gas • wood / charcoal / peat • solar - photovoltaics and heating • wind • waves • hydroelectric • geothermal • nuclear - thermonuclear, fission, and fusion(?) Sources of Energy: Fraction of US Energy supply
1US Energy Information Administration. Sources of Energy: Costs
1Wikipedia, NEA estimates. Sources of Energy: Costs
1Wikipedia, California Energy Commission estimates. Carbon Dioxide
1Graph from http://climate.nasa.gov/evidence/ Global Temperature Record
1Figure by Robert A. Rohde, using data from 11 different studies. Current Event: Larsen C Ice Sheet Breakup Just this week (confirmed yesterday) a 6,000-square-kilometer iceberg (about the size of Delaware) broke off of Antarctica.
1John Sonntag, NASA Current Event: Larsen C Ice Sheet Breakup
1NOAA, NASA CIMSS Collisions
The conservation of momentum is very useful when analyzing what happens to colliding objects.
In all collisions (with no external forces) momentum is conserved:
net momentum before collision = net momentum after collision
pnet,i = pnet,f Inelastic collisions are collisions in which energy is lost as sound, heat, or in deformations of the colliding objects.
When the colliding objects stick together afterwards the collision is perfectly inelastic.
Types of Collision
There are two different types of collisions:
Elastic collisions are collisions in which none of the kinetic energy of the colliding objects is lost. (Ki = Kf ) When the colliding objects stick together afterwards the collision is perfectly inelastic.
Types of Collision
There are two different types of collisions:
Elastic collisions are collisions in which none of the kinetic energy of the colliding objects is lost. (Ki = Kf )
Inelastic collisions are collisions in which energy is lost as sound, heat, or in deformations of the colliding objects. Types of Collision
There are two different types of collisions:
Elastic collisions are collisions in which none of the kinetic energy of the colliding objects is lost. (Ki = Kf )
Inelastic collisions are collisions in which energy is lost as sound, heat, or in deformations of the colliding objects.
When the colliding objects stick together afterwards the collision is perfectly inelastic. 258 Chapter 9 Linear Momentum and Collisions
Because all velocities in Figure 9.7 are either to the left or the right, they can be Before the collision, the particles move separately. represented by the corresponding speeds along with algebraic signs indicating direction. We shall indicate v as positive if a particle moves to the right and nega- S S v1i v2i tive if it moves to the left. 258 Chapter 9 Linear Momentum In a typical and Collisions problem involving elastic collisions, there are two unknown quanti-
Elastic Collisionsm1 m2 ties, and Equations 9.16 and 9.17 can be solved simultaneously to find them. An Because all velocities in Figure 9.7 are either to the left or the right, they can be a Before the collision, the alternative approach, however—one that involves a little mathematical manipula- particles move separately. representedtion of Equation by the 9.17—often corresponding simplifies speeds alongthis process. with algebraic To see how, signs let indicating us cancel the direction.1 We shall indicate v as positive if a particle moves to the right and nega- factor 2 in Equation 9.17 and rewrite it by gathering terms with subscript 1 on the S S tive if it moves to the left. After thev1i collision, thev2 i left and 2 on the right: particles continue to move In a typical problem involving elastic collisions, there are two unknown quanti- 2 2 2 2 separatelym1 with new velocities.m2 ties, and Equations 9.16 and m9.171(v 1cani 2 bev1f solved) 5 m 2simultaneously(v2f 2 v2i ) to find them. An a alternative approach, however—one that involves a little mathematical manipula- S S Factoring both sides of this equation gives v1f v2f tion of Equation 9.17—often simplifies this process. To see how, let us cancel the 1 factor 2 in Equation 9.17 and rewrite it by gathering terms with subscript 1 on the m1(v1i 2 v1f) (v1i 1 v1f) 5 m2(v2f 2 v2i)(v2f 1 v2i) (9.18) After the collision, the left and 2 on the right: b particles continue to move Next, let us separate the terms2 containing2 2 m1 and2 m2 in Equation 9.16 in a simi- separately with new velocities. m1(v1i 2 v1f ) 5 m2(v2f 2 v2i ) Figure 9.7 Schematic represen- lar way to obtain tation ofS an elastic head-onS colli- Factoring both sides of this equation gives v1f v2f sion between two particles. m1(v1i 2 v1f) 5 m2(v2f 2 v2i) (9.19) m1(v1i 2 v1f) (v1i 1 v1f) 5 m2(v2f 2 v2i)(v2f 1 v2i) (9.18) b To obtain our final result, we divide Equation 9.18 by Equation 9.19 and obtain Next, let us separate the terms containing m1 and m2 in Equation 9.16 in a simi- 1 5 1 Figure 9.7 Schematic represen- lar way to obtain v1i v1f v2f v2i tation of an elastic head-on colli- Now rearrange terms once again so as to have initial quantities on the left and final sion between two particles. m1(v1i 2 v1f) 5 m2(v2f 2 v2i) (9.19) quantities on the right: To obtain our final result, we divide Equation 9.18 by Equation 9.19 and obtain v1i 2 v2i 5 2(v1f 2 v2f) (9.20) Pitfall Prevention 9.3 v1i 1 v1f 5 v2f 1 v2i Not a General Equation Equation This equation, in combination with Equation 9.16, can be used to solve problems 9.20 can only be used in a very spe- Nowdealing rearrange with termselastic once collisions. again so This as to pair have of initial equations quantities (Eqs. on 9.16 the leftand and 9.20) final is easier cific situation, a one- dimensional, quantitiesto handle on than the right:the pair of Equations 9.16 and 9.17 because there are no quadratic elastic collision between two terms like there are in Equationv1i 2 v 9.17.2i 5 2According(v1f 2 v2f) to Equation 9.20, the relative(9.20) veloc- Pitfall Prevention 9.3 objects. The general concept is ity of the two particles before the collision, v 2 v , equals the negative of their conservationNot a General of Equation momentum Equation (and This equation, in combination with Equation 9.16,1 ican be2i used to solve problems conservation9.20 can only ofbe kinetic used in energya very spe if - dealingrelative with velocity elastic after collisions. the collision, This pair 2 (ofv1 fequations 2 v2f). (Eqs. 9.16 and 9.20) is easier thecific collision situation, is a elastic) one- dimensional, for an iso- to handleSuppose than the the masses pair of and Equations initial 9.16velocities and 9.17 of bothbecause particles there areare no known. quadratic Equations latedelastic system. collision between two terms9.16 likeand there9.20 arecan in be Equation solved for 9.17. the According final velocities to Equation in terms 9.20, ofthe the relative initial veloc velocities- objects. The general concept is ity of the two particles before the collision, v 2 v , equals the negative of their conservation of momentum (and because there are two equations and two unknowns:1i 2i relative velocity after the collision, 2(v 2 v ). conservation of kinetic energy if m 21f m 2f 2m the collision is elastic) for an iso- Suppose the masses and initial velocities1 2 of both particles2 are known. Equations v1f 5 v1i 1 v2i (9.21) lated system. 9.16 and 9.20 can be solved for them 1final1 m velocities2 inm terms1 1 m 2of the initial velocities because there are two equations and two unknowns: a 2m b am 2 m b 5 1 1 2 1 v2f m1 2 m2 v1i 2m2 v2i (9.22) v 5 m1 1 mv 2 1 m1 1vm 2 (9.21) 1f 1 1i 1 2i m1a m2 b m1 a m2 b It is important to use the appropriatea b signsa for v1i andb v2i in Equations 9.21 and 9.22. 2m1 m2 2 m1 Let us consider somev 5 special cases.v If1 m1 5 m2, Equationsv 9.21 and 9.22(9.22) show that 2f m 1 m 1i m 1 m 2i v1f 5 v2i and v2f 5 v1i , which1 means2 that the1 particles2 exchange velocities if they have equal masses. That isa approximatelyb whata one observesb in head-on billiard ball It is important to use the appropriate signs for v1i and v2i in Equations 9.21 and 9.22. collisions: the cue ball stops and the struck ball moves away from the collision with Let us consider some special cases. If m1 5 m2, Equations 9.21 and 9.22 show that v1thef 5 samev2i and velocity v2f 5 vthe1i , which cue ball means had. that the particles exchange velocities if they have If equal particle masses. 2 is Thatinitially is approximately at rest, then whatv2i 5 one 0, and observes Equations in head-on 9.21 billiardand 9.22 ball become collisions: the cue ball stops and the struck ball moves away from the collision with m1 2 m2 the same velocity the cue ball had. v 5 v (9.23) 1f m 1 m 1i Elastic collision: particle 2 If particle 2 is initially at rest, then v2i 5 0,1 and Equations2 9.21 and 9.22 become initially at rest a b m1 2 m22m v 5 1v (9.23) 1f v2f 5m 1 m 1i v1i (9.24) Elastic collision: particle 2 1 m1 12 m2 initially at rest a b 2am1 b If m1 is much greater than mv2 and5 v2i 5 0, wev see from Equations 9.23 and(9.24) 9.24 that 2f m 1 m 1i v1f < v1i and v2f < 2v1i. That is, when1 a very2 heavy particle collides head-on with a a b If m1 is much greater than m2 and v2i 5 0, we see from Equations 9.23 and 9.24 that v1f < v1i and v2f < 2v1i. That is, when a very heavy particle collides head-on with a The motional energy is just transferred from the first ball to the second!
The first ball comes to rest after the collision, and the second moves off to the right at 1 m/s. Notice that momentum is conserved.
Example: Pool
A pool ball moving at 1 m/s to the right strikes a stationary pool ball in an elastic collision. What is the final velocity of the two pool balls? (Assume they have the same mass.)
1Photo by Max Stansell via hsphysicslab.blogspot.com. Example: Pool
A pool ball moving at 1 m/s to the right strikes a stationary pool ball in an elastic collision. What is the final velocity of the two pool balls? (Assume they have the same mass.)
The motional energy is just transferred from the first ball to the second!
The first ball comes to rest after the collision, and the second moves off to the right at 1 m/s. Notice that momentum is conserved.
1Photo by Max Stansell via hsphysicslab.blogspot.com. Question
A friend is studying collisions and tells you about the result of one of her experiments.
One ball of mass 1 kg and moving initially with a velocity u1 = 5 m/s moving right collides with another of mass 2 kg and velocity u2 = 1 m/s moving left. After the collision the first ball has a velocity of 3 m/s moving left and the second has a velocity of 3 m/s moving right.
Is this collision physical? (Could it really happen?) What is the initial kinetic energy? Is this an elastic collision? Inelastic Collisions
For general inelastic collisions, some kinetic energy is lost.
But we can still use the conservation of momentum:
pi = pf 9.4 Collisions in One Dimension 257
In contrast, the total kinetic energy of the system of particles may or may not be con- served, depending on the type of collision. In fact, collisions are categorized as being either elastic or inelastic depending on whether or not kinetic energy is conserved. An elastic collision between two objects is one in which the total kinetic energy (as well as total momentum) of the system is the same before and after the collision. Collisions between certain objects in the macroscopic world, such as billiard balls, are only approximately elastic because some deformation and loss of kinetic 9.4energy Collisions in One Dimension 257 take place. For example, you can hear a billiard ball collision, so you know that some of the energy is being transferred away from the system by sound. An elastic In contrast, the total kinetic energy of the system of particles may or may not be con- collision mustserved, be perfectly depending silent! on the Truly type ofelastic collision. collisions In fact, occurcollisions between are categorized atomic andas being subatomic particles.either elastic These or inelastic collisions depending are described on whether by or the not isolated kinetic energy system is model conserved. for both energy andAn elastic momentum. collision Furthermore, between two objects there is must one in be which no transformation the total kinetic energy of kinetic energy(as wellinto as other total typesmomentum) of energy of the within system the is thesystem. same before and after the collision. An inelasticCollisions collision between is one certain in which objects the intotal the kineticmacroscopic energy world, of the such system as billiard is not balls, the same beforeare only and approximately after the collision elastic because (even thoughsome deformation the momentum and loss of of the kinetic system energy is conserved).take Inelastic place. For collisions example, are you of cantwo hear types. a billiardWhen the ball objects collision, stick so youtogether know that Pitfall Prevention 9.2 after they collide,some of as the happens energy whenis being a meteoritetransferred collides away from with the the system Earth, by thesound. collision An elastic Inelastic Collisions Generally, inelastic collisions are hard to is called perfectlycollision inelastic. must be perfectly When thesilent! colliding Truly elastic objects collisions do not occur stick between together atomic but and subatomic particles. These collisions are described by the isolated system model for analyze without additional infor- some kineticboth energy energy is transformed and momentum. or transferred Furthermore, away, there as mustin the be case no transformation of a rubber of mation. Lack of this information ball collidingkinetic with energya hard into surface, other typesthe collision of energy is within called the inelastic system. (with no modify- appears in the mathematical representation as having more ing adverb). WhenAn inelastic the rubber collision ball is collides one in which with thethe totalhard kinetic surface, energy some of of the the system ball’s is not unknowns than equations. kinetic energythe issame transformed before and whenafter thethe collision ball is deformed(even though while the itmomentum is in contact of the with system the surface.is Inelastic conserved). collisions Inelastic are collisions described are ofby twothe types.momentum When the version objects of stick the togetheriso- Pitfall Prevention 9.2 after they collide, as happens when a meteorite collides with the Earth, the collision Inelastic Collisions Generally, lated system model. The system could be isolated for energy, with kinetic energy inelastic collisions are hard to transformedis to called potential perfectly or internal inelastic. energy. When Ifthe the colliding system isobjects nonisolated, do not stick there together could but analyze without additional infor- be energy leavingsome kinetic the system energy byis transformedsome means. or Intransferred this latter away, case, as inthere the casecould of alsoa rubber mation. Lack of this information ball colliding with a hard surface, the collision is called inelastic (with no modify- appears in the mathematical be some transformation of energy within the system. In either of these cases, the representation as having more kinetic energying ofadverb). the system When changes. the rubber ball collides with the hard surface, some of the ball’s kinetic energy is transformed when the ball is deformed while it is in contact with unknowns than equations. In the remainderthe surface. of Inelasticthis section, collisions we investigate are described the by mathematical the momentum details version for of col the- iso- lisions in onelated dimension system model. and The consider system the could two be extreme isolated cases,for energy, perfectly with kinetic inelastic energy and elastic collisions.transformed to potential or internal energy. If the system is nonisolated, there could be energy leaving the system by some means. In this latter case, there could also Perfectly beInelastic some transformation Collisions of energy within the system. In either of these cases, the kinetic energy of the system changes. S S Consider two particlesIn the remainder of masses of mthis1 and section, m2 moving we investigate with initial the mathematical velocities v 1detailsi and vfor2i col- along the samelisions straight in one line dimension as shown and in consider Figure 9.6. the The two extremetwo particles cases, collide perfectly head- inelastic S on, stick together,and elastic and collisions. then move with some common velocity v f after the collision. Because the momentum of an isolated system is conserved in any collision, we can Before the collision, the say that the Perfectlytotal momentum Inelastic before Collisions the collision equals the total momentum of the particles move separately. composite system after the collision: S S Consider two particles of masses m1 and m2 moving with initial velocities v 1i and v 2i S S S S S S alongDp 5 the 0 same S straightpi 5 pf line S as mshown1 v 1i 1in mFigure2 v 2i 5 9.6.m The1 1 twom2 particlesv f collide(9.14) head- S S S on, stick together, and then move with some common velocity v f after the collision. v1i v2i Solving for the final velocity gives m1 m Because the momentum of an isolated system is conserved1 in2 any collision, we can Before the collision, the2 S 1 S Perfectly Inelastic Collisions say that the total momentumS m before1 v 1i them 2collisionv 2i equals the total momentum of the a particles move separately. composite system afterv thef 5 collision: (9.15) m1 1 m2 S S S S S S Dp 5 0 S pi 5 pf S m1 v 1i 1 m2 v 2i 5 m1 1 m2 v f (9.14) After theS collision,S the v1i v2i Solving for the final velocity gives m1 Elastic Collisions 1 2 particles move together.m2 S S m1 v 1i 1 m2 v 2i S S a Consider two particles of masses m1 and mS2 moving with initial velocities v 1i and v 2i v f 5 (9.15) along the same straight line as shown in Figure m9.71 1 onm 2page 258. The two particles S S collide head-on and then leave the collision site with different velocities, v 1f and After the collision,v fthe S m ϩ m v2f . In an elasticElastic collision, Collisions both the momentum and kinetic energy of the system particles1 move2 together. S S are conserved.Consider Therefore, two particles considering of masses velocities m1 and m along2 moving the with horizontal initial velocities direction v 1i andin v 2i b Figure 9.7, wealong have the same straight line as shown in Figure 9.7 on page 258. The two particlesNow the two particles stick together after colliding ⇒ same final velocity!S S collide head-on and then leave the collision site with different velocities, v 1f and Figure 9.6 Schematic reprevf - S p 5 p S m v 1 m v 5 m v 1 m v (9.16) m ϩ m v2f . Ini an elasticf collision,1 1i both2 2i the momentum1 1f 2 2 fand kinetic energy of the system sentation of a perfectly1 2 inelastic head-onpi = pf collision⇒ m1v 1betweeni + m2v2i two= (m 1 + m2)vf are conserved. Therefore,1 2 considering1 2 velocities1 2along1 the horizontal2 direction in b FigureKi 59.7, K wef haveS 2m1v1i 1 2m2v2i 5 2m1v1f 1 2m2v2f (9.17) particles. Figure 9.6 Schematic repre- pi 5 pf S m1v1i 1 m2v2i 5 m1v1f 1 m2v2f (9.16) sentation of a perfectly inelastic head-on collision between two 1 2 1 2 1 2 1 2 Ki 5 Kf S 2m1v1i 1 2m2v2i 5 2m1v1f 1 2m2v2f (9.17) particles. pnet,i = pnet,f
mvi = (2m)vf
10m = 2mvf
The final mass is twice as much, so the final speed must be only half as much: vf = 5 m/s.
Inelastic Collision Example
From page 91-92 of Hewitt:
Two freight rail cars collide and lock together. Initially, one is moving at 10 m/s and the other is at rest. Both have the same mass. What is their final velocity? Inelastic Collision Example
From page 91-92 of Hewitt:
Two freight rail cars collide and lock together. Initially, one is moving at 10 m/s and the other is at rest. Both have the same mass. What is their final velocity?
pnet,i = pnet,f
mvi = (2m)vf
10m = 2mvf
The final mass is twice as much, so the final speed must be only half as much: vf = 5 m/s. Collision Question
Two objects collide and move apart after the collision. Could the collision be inelastic?
(A) Yes. (B) No. Collision Question
Two objects collide and move apart after the collision. Could the collision be inelastic?
(A) Yes. ← (B) No. Question
In a perfectly inelastic one-dimensional collision between two moving objects, what condition alone is necessary so that the final kinetic energy of the system is zero after the collision?
(A) The objects must have initial momenta with the same magnitude but opposite directions. (B) The objects must have the same mass. (C) The objects must have the same initial velocity. (D) The objects must have the same initial speed, with velocity vectors in opposite directions.
1Serway & Jewett, page 259, Quick Quiz 9.5. Question
In a perfectly inelastic one-dimensional collision between two moving objects, what condition alone is necessary so that the final kinetic energy of the system is zero after the collision?
(A) The objects must have initial momenta with the same magnitude but opposite directions. ← (B) The objects must have the same mass. (C) The objects must have the same initial velocity. (D) The objects must have the same initial speed, with velocity vectors in opposite directions.
1Serway & Jewett, page 259, Quick Quiz 9.5. Summary • sources of energy • rotational motion
Essay Homework due July 19th. • Describe the design features of cars that make them safer for passengers in collisions. Comment on how the design of cars has changed over time to improve these features. In what other circumstances might people be involved in collisions? What is / can be done to make those collisions safer for the people involved? Make sure use physics principles (momentum, impulse) in your answers! Homework Hewitt, • Ch 6, onward from page 96, Plug and chug: 1, 3, 5, 7; Ranking: 1; Exercises: 5, 7, 19, 31, 47 • Ch 7, onward from page 119. Exercises: 57, 59; Probs: 7, 9