Momentum. Momentum. Momentum. January 25, momentum presentation Table of Contents. Momentum Defined. Grade:«grade»

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1 oentu presentation 2016 New Jersey Center for Teaching and Learning Progressive Science Initiative This aterial is ade freely available at wwwnjctlorg and is intended for the non coercial use of students and teachers These aterials ay not be used for any coercial purpose without the written perission of the owners NJCTL aintains its website for the convenience of teachers who wish to ake their work available to other teachers, participate in a virtual professional learning counity, and/or provide access to course aterials to parents, students and others Moentu wwwnjctlorg Click to go to website: wwwnjctlorg wwwnjctlorg wwwnjctlorg Table of Contents Click on the topic to go to that section Moentu Ipulse Moentu Equation The Moentu of a Syste of Objects Conservation of Moentu Types of Collisions Collisions in Two Diensions Moentu Return to Table of Contents Table of Contents Moentu Moentu Moentu Defined Newton s First Law tells us that objects reain in otion with a constant velocity unless acted upon by an external force (Law of Inertia) In our experience: Grade:«grade» Subject:AP Physics 1 Date:«date» When objects of different ass travel with the sae velocity, the one with ore ass is harder to stop When objects of equal ass travel with different velocities, the faster one is harder to stop May 11 4:02 PM Moentu 1

2 oentu presentation 2016 Moentu Defined Define a new quantity, oentu (p), that takes these observations into account: oentu = ass velocity p =v Since: ass is a scalar quantity velocity is a vector quantity the product of a scalar and a vector is a vector and: oentu = ass velocity therefore: Moentu is a Vector Quantity Moentu is a vector quantity it has agnitude and direction click here for a introductory video on oentu fro Bill Nye! Moentu Moentu is a vector SI Unit for Moentu There is no specially naed unit for oentu so there is an opportunity for it to be naed after a renowned physicist! We use the product of the units of ass and velocity 1 Which has ore oentu? A A large truck oving at 30 /s B A sall car oving at 30 /s C Both have the sae oentu ass x velocity kg /s Moentu unit Moentu proble 2 What is the oentu of a 20 kg object oving to the right with a velocity of 5 /s? 3 What is the oentu of a 20 kg object with a velocity of 50 /s to the left? Moentu proble Moentu proble 2

3 oentu presentation What is the velocity of a 5 kg object whose oentu is 15 kg /s? 5 What is the ass of an object that has a oentu of 35 kg /s with a velocity of 7 /s? Moentu proble Nov 25 5:20 PM Change in Moentu Suppose that there is an event that changes an object's oentu Ipulse Moentu Equation fro p 0 the initial oentu (just before the event) by Δp the change in oentu to p f the final oentu (just after the event) Return to Table of Contents The equation for oentu change is: Ipulse Moentu Equation Moentu change Moentu Change = Ipulse Moentu Change = Ipulse Moentu change equation: Let's now use Newton's Second Law, and see if we can relate Δp to the external force in an explicit anner: Newton's First Law tells us that the velocity (and so the oentu) of an object won't change unless the object is affected by an external force Look at the above equation Can you relate Newton's First Law to the Δp ter? Δp would represent the external force We've found that the oentu change of an object is equal to an external Force applied to the object over a period of tie Ipulse Moentu equation Ipulse Moentu equation 3

4 oentu presentation 2016 Ipulse Moentu Equation The force acting over a period of tie on an object is now defined as Ipulse and we have the Ipulse Moentu equation: SI Unit for Ipulse There no special unit for ipulse We use the product of the units of force and tie force x tie N s Recall that N=kg /s 2, so N s=kg /s 2 x s = kg /s This is also the unit for oentu, which is a good thing since Ipulse is the change in oentu Ipulse Moentu equation Ipulse unit 6 There is a battery powered wheeled cart oving towards you at a constant velocity You want to apply a force to the cart to ove it in the opposite direction Which cobination of the following variables will result in the greatest change of oentu for the cart? Select two answers 7 Fro which law or principle is the Ipulse Moentu equation derived fro? A Conservation of Energy A Increase the applied force B Increase the tie that the force is applied C Maintain the sae applied force B Newton's First Law C Newton's Second Law D Conservation of Moentu D Decrease the tie that the force is applied Ipulse Moentu equation proble Ipulse Moentu equation proble Can the ipulse applied to an object be negative? Why or why not? Give an exaple to explain your answer Effect of Collision Tie on Force Ipulse = F( t) = F( t) F (newtons) Since force is inversely proportional to Δt, changing the t of a given ipulse by a sall aount can greatly change the force exerted on an object! t (seconds) Ipulse Moentu equation proble Ipulse and tie 4

5 oentu presentation 2016 Every Day Applications Ipulse = F( t) = F( t) Every Day Applications I=FΔt=Δp The inverse relationship of Force and tie interval leads to any interesting applications of the Ipulse Moentu equation to everyday experiences such as: car structural safety design car air bags landing after parachuting artial arts hitting a baseball catching a basebal Let's analyze two specific cases fro the previous list: car air bags hitting a baseball Whenever you have an equation like the one above, you have to decide which values will be fixed and which will be varied to deterine the ipact on the third value For the car air bags, we'll fix Δp, vary Δt and see its ipact on F For the bat hitting a ball, we'll fix F, vary Δt and see the ipact on Δp Every Day Applications Every Day Applications Car Air Bags I=FΔt=Δp In the Dynaics unit of this course, it was shown how during an accident, seat belts protect passengers fro the effect of Newton's First Law by stopping the passenger with the car, and preventing the fro striking the dashboard and window They also provide another benefit explained by the Ipulse Moentu equation But, this benefit is greatly enhanced by the presence of air bags Can you see what this benefit is? Car Air Bags I=FΔt=Δp The seat belt also increases the tie interval that it takes the passenger to slow down there is soe play in the seat belt, that allows you to ove forward a bit before it stops you The Air bag will increase that tie interval uch ore than the seat belt by rapidly expanding, letting the passenger strike it, then deflating Earlier it was stated that for the Air bag exaple, Δp would be fixed and Δt would be varied So, we've just increased Δt Why is Δp fixed? Car Air Bags Car Air Bags Car Air Bags I=FΔt=Δp Δp is fixed, because as long as the passenger reains in the car, the car (and the passengers) started with a certain velocity and finished with a final velocity of zero, independent of seat belts or air bags Rearranging the equation, we have: F represents the Force delivered to the passenger due to the accident Car Air Bags I=FΔt=Δp Since Δp is fixed, by extending the tie interval (Δt increases) that it takes a passenger to coe to rest (seat belt and air bag), the force, F delivered to the passenger is saller Less force on the passenger eans less physical har Also, another benefit needs a quick discussion of Pressure Pressure is Force per unit area By increasing the area of the body that feels the force (the air bag is big), less pressure is delivered to parts of the body reducing the chance of puncturing the body Also good Car Air Bags Car Air Bags 5

6 oentu presentation If a car is in a front end collision, which of the below factors will help reduce the injury to the driver and passengers? Select two answers A An absolutely rigid car body that doesn't defor B Deployent of an air bag for each adult in the car 9 Which of the following variables are fixed when a oving car strikes a barrier and coes to rest? A Force delivered to the car B Interval of tie for the collision C Deployent of an air bag only for the driver D The proper wearing of a seatbelt or child seat for each person in the car C Moentu change D Acceleration of the car Car Air Bags proble Car Air Bags proble Hitting a Baseball Hitting a Baseball I=FΔt=Δp I=FΔt=Δp Now, we're going to take a case where there is a given force, and the tie interval will be varied to find out what happens to the change of oentu This is different fro the Air Bag exaple just worked (Δp was constant, Δt was varied, and its ipact on F was found) Assue a baseball batter swings with a given force that is applied over the interval of the ball in contact with the bat The ball is approaching the batter with a positive oentu The batter wants to hit the ball and get the largest Δp possible for his force which depends on his strength and tiing The greater the Δp, the faster the ball will fly off his bat, which will result in it going further, hopefully to the seats Hitting a baseball is way ore coplex than the analysis that will follow If you're interested in ore inforation, please check out the book, The Physics of Baseball, written by Robert Adair, a Yale University physicist What is the goal of the batter? Hitting a Baseball Hitting a Baseball Hitting a Baseball Every Day Applications I=FΔt=Δp I=FΔt=Δp In this case, the batter wants to axiize the tie that his bat (which is providing the force) is in contact with the ball This eans he should follow through with his swing Now, discuss the other exaples Make sure you decide which object in the collision is ore "affected" by the force or the change in oentu, and which variables are capable of being varied The batter needs to apply a large ipulse to reverse the ball's large oentu fro the positive direction (towards the batter) to the negative direction (heading towards the center field bleachers) Consider the Air Bag exaple the car experiences the sae ipulse as the passenger during an accident, but a car is less valuable than a huan being so it is ore iportant for the passenger that less force is delivered to his body and ore force is absorbed by the car car structural safety design landing after parachuting artial arts catching a basebal Hitting a Baseball Every Day Applications 6

7 oentu presentation An external force of 25 N acts on a syste for 100 s What is the agnitude of the ipulse delivered to the syste? 11 In the previous proble, an external force of 25 N acted on a syste for 100 s The ipulse delivered was 250 N s What was the change in oentu of the syste? Ipulse proble Ipulse proble 12 The oentu change of an object is equal to the A B force acting on it ipulse acting on it C velocity change of the object D object's ass ties the force acting on it 13 Air bags are used in cars because they: A increase the force with which a passenger hits the dashboard B increase the duration (tie) of the passenger's ipact C decrease the oentu of a collision D decrease the ipulse in a collision B Ipulse proble Ipulse proble 14 One car crashes into a concrete barrier Another car crashes into a collapsible barrier at the sae speed What is the difference between the two crashes? Select two answers 15 In order to increase the final oentu of a golf ball, the golfer should: Select two answers: A B aintain the speed of the golf club after the ipact (follow through) Hit the ball with a greater force A change in oentu C Decrease the tie of contact between the club and the ball B force on the car C ipact tie D Decrease the initial oentu of the golf club D final oentu Ipulse proble May 12 10:24 AM 7

8 oentu presentation An external force acts on an object for s During that tie the object's oentu increases by 400 kg /s What was the agnitude of the force? 17 A 50,000 N force acts for 0030 s on a 25 kg object that was initially at rest What is its final velocity? Ipulse proble Ipulse proble 18 A 1200 kg car slows fro 400 /s to 50 /s in 20 s What force was applied by the brakes to slow the car? Graphical Interpretation of Ipulse Graphs are a very valuable ethod to solve physics probles So far, we've dealt with a constant force exerted over a given tie interval But forces are not always constant ost of the tie, they are changing over tie In the baseball exaple, the force of the bat starts out very sall as it akes initial contact with the ball It then rapidly rises to its axiu value, then decreases as the ball leaves the bat This would be a very difficult proble to handle without a graph Start with representing a constant force over a tie interval, Δt, and plot Force vs tie on a graph Ipulse proble Ipulse graph Graphical Interpretation of Ipulse F [N] Graphical Interpretation of Ipulse F [N] F Area = F Area = Ipulse (Change in Moentu) Ipulse (Change in Moentu) t t [s] t t [s] Note, the area under the Force Tie graph is: height base = F t = I (ipulse) = p (change in oentu) By taking the area under a Force tie graph, we have found both Ipulse and Δp If the force is not constant, the area can be found by breaking it into solvable shapes (rectangles, triangles) and suing the up Ipulse graph Ipulse graph 8

9 oentu presentation Using the F t graph shown, what is the change in oentu during the tie interval fro 0 to 6 s? 20 A 5 kg object with an initial velocity of 3 /s experiences the force shown in the graph What is its velocity at 6 s? Ipulse graph proble Ipulse graph proble 21 Using the F t graph shown, what is the change in oentu during the tie interval fro 6 to 10 s? The Moentu of a Syste of Objects Return to Table of Contents Ipulse graph proble Moentu of a Syste Moentu of a Syste Grade:«grade» Subject:AP Physics 1 Date:«date» The Moentu of a Syste of Objects If a syste contains ore than one object, its total oentu is the vector su of the oenta of those objects p syste = p p syste = p 1 + p 2 + p 3 + p syste = 1v 1 + 2v 2 + 3v 3 + It's critically iportant to note that oenta add as vectors, not as scalars May 12 11:34 AM Moentu of a Syste 9

10 oentu presentation 2016 The Moentu of a Syste of Objects Exaple psyste = 1v1 + 2v2 + 3v3 + Deterine the oentu of a syste of two objects: 1, has a ass of 6 kg and a velocity of 13 /s towards the east and 2, has a ass of 14 kg and a velocity of 7 /s towards the west In order to deterine the total oentu of a syste: + 13 /s 1 = 6 kg v1 = 13 /s 7 /s East (+) 2 = 14 kg v2 = 7 /s psyste = p1 + p2 psyste = 1v1 + 2v2 Then: Add the oenta to get a total oentu of the syste Moentu of a Syste psyste = 1v1 + 2v2 = (6 kg)(13 /s) + (14 kg)( 7 /s) = (78 kg /s) + ( 98 kg /s) = 20 kg /s psyste = 1v1 + = (6 kg)(13 = (78 kg /s = 20 kg / Moentu of a Syste Deterine the oentu of a syste of two objects: 1, has a ass of 60 kg and a velocity of 20 /s north and 2, has a ass of 3 kg and a velocity 20 /s south kg 6 kg Deterine the oentu of a syste of two objects: the first has a ass of 80 kg and a velocity of 80 /s to the east while the second has a ass of 50 kg and a velocity of 15 /s to the west 1 = 6 kg v1 = +20 /s (North) 1 = 8 kg v1 = +8 /s (E 2 = 3 kg v2 = 20 /s (South) 2 = 5 kg v2 = 15 /s First: Deterine a direction to be considered positive Assign positive values to oenta in that direction Assign negative values to oenta in the opposite direction psyste = 1v1 + 2v2 = (6 kg)(20 /s) + (3 kg)( 20 /s) = (120 kg /s) + ( 60 kg /s) = 60 kg /s (agnitude) psyste = 1v1 + = (8 kg)(8 = (64 kg /s = 11 kg /s direction is North Moentu of a Syste proble Deterine the oentu of a syste of three objects: The first has a ass of 70 kg and a velocity of 23 /s north the second has a ass of 90 kg and a velocity of 7 /s north and the third has a ass of 50 kg and a velocity of 42 /s south Conservation of Moentu 3 Moentu of a Syste proble psyste = 1v1 + 2v2 + 3v3 = (7kg)(23/s) + (9kg)(7/s) +(5 kg)( 42/s) = (161 kg /s) + (63 kg /s) + ( 210 kg /s) Return to Table of Contents = 14 kg /s (North) Moentu of a Syste proble Conservation of Moentu 10

11 oentu presentation 2016 Conservation Laws Moentu is Conserved Soe of the ost powerful concepts in science are called "conservation laws" This was covered in detail in the Energy unit please refer back to that for ore detail Here is a suary In the last unit we learned that energy is conserved Conservation laws: Moentu is not created or destroyed Like energy, oentu is a conserved property of nature This eans: apply to closed systes where the objects only interact with each other and nothing else The total oentu in a closed syste is always the sae enable us to solve probles without worrying about the details of an event The only way the oentu of a syste can change is if oentu is added or taken away by an outside force Conservation of Moentu Conservation of Moentu Conservation of Moentu Conservation of Moentu We will use the Conservation of Moentu to explain and predict the otion of a syste of objects The oentu of an object changes when it experiences an ipulse (I=FΔt): As with energy, it will only be necessary to copare the syste at two ties: just before and just after an event This ipulse arises when a non zero external force acts on the object This is exactly the sae for a syste of objects Conservation of Moentu Conservation of Moentu Conservation of Moentu Why don't the internal forces on a syste change the oentu of the syste? If there is no net external force on the syste, then F = 0, and since I = FΔt, I is also zero: Since psyste, 0 = psyste, f, the oentu of the syste is conserved, that is, it does not change Conservation of Moentu The internal for interaction of th syste By Ne these are all ac and viewed fro all cancel out force acting on oentu is c Conservation of Moentu proble 11

12 oentu presentation An external, positive force acts on a syste of objects Which of the following are true? Select two answers A The velocity of the syste reains the sae B The velocity of the syste increases Types of Collisions C The oentu of the syste decreases D The oentu of the syste increases Return to Table of Contents Conservation of Moentu proble Types of Collisions Types of Collisions Objects in an isolated syste can interact with each other in two basic ways: They can collide If they are stuck together, they can explode (push apart) In an isolated syste both oentu and total energy are conserved But the energy can change fro one for to another Conservation of oentu and change in kinetic energy can help predict what will happen in these events Types of Collisions We differentiate collisions and explosions by the way the energy changes or does not change for inelastic collisions: two objects collide, converting soe kinetic energy into other fors of energy such as potential energy, heat or sound elastic collisions: two objects collide and bounce off each other while conserving kinetic energy energy is not transfored into any other type explosions: an object or objects breaks apart because potential energy stored in one or ore of the objects is transfored into kinetic energy Types of Collisions Types of Collisions Inelastic Collisions There are two types of Inelastic Collisions perfect inelastic collisions: two objects collide, stick together and ove as one ass after the collision, transferring kinetic energy into other fors of energy general inelastic collisions: two objects collide and bounce off each other, transferring kinetic energy into other fors of energy There is really no such thing as a perfect elastic collision During all collisions, soe kinetic energy is always transfored into other fors of energy But soe collisions transfor so little energy away fro kinetic energy that they can be dealt with as perfect elastic collisions In cheistry, the collisions between olecules and atos are odeled as perfect elastic collisions to derive the Ideal Gas Law Other exaples include a steel ball bearing dropping on a steel plate, a rubber "superball" bouncing on the ground, and billiard balls bouncing off each other Inelastic Collsions 12

13 oentu presentation 2016 Explosions Explosions A firecracker is an exaple of an explosion The cheical potential energy inside the firecracker is transfored into kinetic energy, light and sound A cart with a copressed spring is a good exaple When the spring is against a wall, and it is released, the cart starts oving converting elastic potential energy into kinetic energy and sound Think for a oent can you see a reseblance between this phenoenon and either an elastic or inelastic collision? In both an inelastic collision and an explosion, kinetic energy is transfored into other fors of energy such as potential energy But they are tie reversed! An inelastic collision transfors kinetic energy into other fors of energy, such as potential energy An explosion changes potential energy into kinetic energy Thus, the equations to predict their otion will be inverted The next slide suarizes the four types of collisions and explosions Explosions Explosions Collisions and Explosions Event Description Moentu Conserved? General Inelastic Collision Perfect Inelastic Collision Elastic Collision Explosion Objects bounce off each other Objects stick together Objects bounce off each other Object or objects break apart Yes Yes Yes Yes Kinetic Energy Conserved? No Kinetic energy is converted to other fors of energy No Kinetic energy is converted to other fors of energy Yes No Release of potential energy increases kinetic energy 5 In collisions oentu is conserved A Elastic B Inelastic C All Types of Collisions Types of Collisions proble 6 In collisions kinetic energy is conserved A Elastic B Inelastic C All 7 In an inelastic collision, kinetic energy is transfored into what after the collision? Select two answers A Nothing, kinetic energy is conserved B More kinetic energy C Theral energy D Light energy Types of Collisions proble Types of Collisions proble 13

14 oentu presentation 2016 Conservation of Moentu During a collision or an explosion, easureents show that the total oentu of a closed syste does not change The diagra below shows the objects approaching, colliding and then separating Perfect Inelastic Collisions In a perfect inelastic collision, two objects collide and stick together, oving afterwards as one object A Av A Bv B B +x the prie eans "after" Before (oving towards each other) p A= Av A p B= Bv B After (oving together) p A'+p B'=( A+ B)v' A B A B A B Av A' A B Bv B' If the easureents don't show that the oentu is conserved, then this would not be a valid law Fortunately they do, and it is! Conservation of Moentu Perfect Inelastic Collisions 8 A 13,500 kg railroad freight car travels on a level track at a speed of 45 /s It collides and couples with a 25,000 kg second car, initially at rest and with brakes released No external force acts on the syste What is the speed of the two cars after colliding? 9 A cannon ball with a ass of 1000 kg flies in horizontal direction with a speed of 250 /s and strikes a ship initially at rest The ass of the ship is 15,000 kg Find the speed of the ship after the ball becoes ebedded in it Perfect Inelastic Collisions proble Perfect Inelastic Collisions proble 10 A 40 kg girl skates at 55 /s on ice toward her 70 kg friend who is standing still, with open ars As they collide and hold each other, what is their speed after the collision? Explosions In an explosion, one object breaks apart into two or ore pieces (or coupled objects break apart), oving afterwards as separate objects To ake the probles solvable at this ath level, we will assue: the object (or a coupled pair of objects) breaks into two pieces the explosion is along the sae line as the initial velocity Perfect Inelastic Collisions proble Explosion 14

15 oentu presentation 2016 Explosions and 2D Explosions Here's a scheatic of an explosion Can you see how it's the tie reversed picture of a perfectly inelastic collision? Grade:«grade» Subject:AP Physics 1 Date:«date» Before (oving together) p A+p B=( A+ B)v A B After (oving apart) p A'= Av A' p B'= Bv B' A B May 12 11:39 AM Explosion 1 A 5 kg cannon ball is loaded into a 300 kg cannon When the cannon is fired, it recoils at 5 /s What is the cannon ball's velocity after the explosion? 2 Two railcars, one with a ass of 4000 kg and the other with a ass of 6000 kg, are at rest and stuck together To separate the a sall explosive is set off between the The 4000 kg car is easured travelling at 6 /s How fast is the 6000 kg car going? Explosion proble Explosion proble In an elastic collision, two objects collide and bounce off each other, as shown below, and both oentu and kinetic energy are conserved This will give us two siultaneous equations to solve to predict their otion after the collision Before (oving towards) After (oving apart) p A= Av A p B= Bv B p A'= Av A' p B'= Bv B' A B A B Before (oving towards) After (oving apart) p A= Av A p B= Bv B p A'= Av A' p B'= Bv B' A B A B 15

16 oentu presentation 2016 Elastic Collision Siultaneous Equations Since both of these equations describe the otion of the sae syste, they ust both be true hence they can be solved as siultaneous equations That will be done on the next slide, and an interesting result will appear Conservation of Moentu 1v 1 + 2v 2 = 1v 1' + 2v 2' 1v 1 1v 1' = 2v 2' 2v 2 1(v 1 v 1') = 2(v 2' v 2) Conservation of Kinetic Energy ½ 1v ½ 2v 2 2 = ½ 1v 1' 2 +½ 2v 2' 2 1v v 2 2 = 1v 1' 2 + 2v 2' 2 1v 2 1 1v 1' 2 = 2v 2' 2 2v 2 2 1(v 2 1 v 1' 2 ) = 2(v 2' 2 v 22 ) 1(v 1 + v 1')(v 1 v 1') = 2(v 2' + v 2)(v 2' v 2) 1(v 1 + v 1')(v 1 v 1') = 2(v 2' + v 2)(v 2' v 2) 1(v 1 v 1') = 2(v 2' v 2) v 1 + v 1' = v 2' + v 2 v 1 v 2 = (v 1' v' 2) Properties of By solving the conservation of oentu and constant kinetic energy equations siultaneously, the following result appeared: 3 Two objects have an elastic collision Before they collide, they are approaching with a velocity of 4 /s relative to each other With what velocity do they ove apart fro one another after the collision? v 1 v 2 = (v 1' v' 2) Do you recognize the ters on the left and right of the above equation? And, what does it ean? The ters are the relative velocities of the two objects before and after the collision It eans that for all elastic collisions regardless of ass the relative velocity of the objects is the sae before and after the collision proble 4 Two objects have an elastic collision Object 1, has an initial velocity of +40 /s and 2 has a velocity of 30 /s After the collision, 1 has a velocity of 10 /s What is the velocity of 2? 5 Two objects have an elastic collision Object 1, has an initial velocity of +60 /s and 2 has a velocity of 20 /s After the collision, 1 has a velocity of 10 /s What is the velocity of 2? proble proble 16

17 oentu presentation 2016 Properties of Properties of Using the conservation of kinetic energy and the conservation of oentu equations, and solving the in a slightly different anner (the algebra is a little ore intense), the following equations result: Using these equations, we will predict the otion of three specific cases, and relate the to observation: Two sae ass particles collide A very heavy particle collides with a light particle at rest A light particle collides head on with a heavy particle at rest Collision of sae ass particles Collision of sae ass particles For a collision of two particles of the sae ass, = 1 = 2 When this substitution is ade into the two equations on the left (try it!), they siplify to: v 1' = v 2 and v 2' = v 1 v 1' = v 2 and v 2' = v 1 The particles exchange velocities A good exaple of this is playing billiards The cue ball and the striped and solid balls are all the sae ass When the cue ball hits a solid ball, the cue ball stops, and the solid ball takes off with the cue ball's velocity v'1 = v2 v'2 = v1 v'1 = v2 v'2 = v1 v1 v2 v1 v2 Collision of sae ass particles Collision of sae ass particles v 1' = v 2 and v 2' = v 1 v 1' = v 2 and v 2' = v 1 The particles exchange velocities Another good exaple is found in nuclear reactor engineering When Uraniu 235 fissions, it releases 3 high speed neutrons These neutrons are oving too fast to cause another Uraniu 235 nucleus to fission, so they have to be slowed down What type of ato would be of ost use here? v'1 = v2 v'2 = v1 The particles exchange velocities An ato with a siilar ass to neutrons A Hydrogen ato oving slower than the neutrons would be great the neutron would exchange velocities, slowing down, while speeding up the hydrogen ato Operational reactors use water for this purpose (heavier than hydrogen, but has any other benefits nuclear engineering is not as siple as elastic collisions!) v'1 = v2 v'2 = v1 v1 v2 v1 v2 17

18 oentu presentation 2016 Large ass striking lighter ass at rest For this case, 1 >> 2 (>> is the sybol for uch greater), which eans that 1 ± 2 1 The equations siplify to: Large ass striking lighter ass at rest v 1' = v 1 and v 2' = 2v 1 1 v 1' = v 1 and v 2' = 2v 1 1 The large ass particle continues on its way with a slightly saller velocity The sall particle oves at twice the initial speed of the large ass particle Picture a bowled bowling ball striking a beach ball at rest v1 v2 = 0 v'1 v1 v'2 = 2v1 2 2 v1 v2 = 0 v'1 v1 v'2 = 2v1 Sall ass striking heavier ass at rest For this case, 1 << 2 (<< is the sybol for uch less than), which eans that 1 ± 2 2 The equations siplify to: v 1 ' = v 1 and v 2 ' 0 Sall ass striking heavier ass at rest v 1 ' = v 1 and v 2 ' 0 The large ass particle reains ostly at rest The sall particle reverses direction, with the sae agnitude as its initial velocity Picture a oving golf ball hitting a stationary bowling ball v v'1 1 v1 1 v2 = 0 v'2 = 0 v2 = 0 v'1 v'2 = 0 6 A bowling ball has a velocity of +v when it collides with a ping pong ball that is at rest The velocity of the bowling ball is virtually unaffected by the collision What will be the speed of the ping pong ball? 7 A baseball bat has a velocity of +v when it collides with a baseball that has a velocity of 2v The bat barely changes velocity during the collision How fast is the baseball going after it's hit? proble proble 18

19 oentu presentation Two objects with identical asses have an elastic collision: the initial velocity of 1 is +60 /s and 2 is 30 /s What is the velocity of 1 after the collision? 9 Two objects with identical asses have an elastic collision: the initial velocity of 1 is +60 /s and 2 is 30 /s What is the velocity of 2 after the collision? proble proble 10 A golf ball is hit against a solid ceent wall, and experiences an elastic collsion The golf ball strikes the wall with a velocity of +35 /s What velocity does it rebound with? Collisions in Two Diensions Return to Table of Contents proble Collisions in Two Diensions 2D collisions Grade:«grade» Subject:AP Physics 1 Date:«date» Conservation of Moentu in Two Diensions Moentu vectors (like all vectors) can be expressed in ters of coponent vectors relative to a reference frae This eans that the oentu conservation equation p = p' can be solved independently for each coponent: This, of course also applies to three diensions, but we'll stick with two for this chapter! May 12 11:40 AM Collisions in Two Diensions 19

20 oentu presentation 2016 Exaple: Collision with a Wall Consider the case of a golf ball colliding elastically with a hard wall, rebounding with the sae velocity, where its angle of incidence equals its angle of reflection Is oentu conserved in this proble? p y' p y p x' p x p' The solid lines represent the oentu of the ball (blue prior to collision, red after the collision) The dashed lines are the x and y coponents of the oentu vectors p Exaple: Collision with a Wall Moentu is not conserved! An external force fro the wall is being applied to the ball in order to reverse its direction in the x axis However, since we have an elastic collision, the ball bounces off the wall with the sae speed that it struck the wall Hence, the agnitude of the initial oentu and the final oentu is equal: p y' p y p x' Now it's tie to resolve oentu into coponents along the x and y axis p x p' p Collisions in Two Diensions Collisions in Two Diensions Exaple: Collision with a Wall Apply the Ipulse Moentu Theore in two diensions: What does this tell us? p y ' p y p x ' p x p' p Exaple: Collision with a Wall Moentu is conserved in the y direction, but it is not conserved in the x direction The initial and final oentu in the y direction are the sae The initial and final oentu in the x direction are equal in agnitude, but opposite in direction they reverse p y ' p y p x ' p x p' p Collisions in Two Diensions Collisions in Two Diensions 1 A tennis ball of ass strikes a wall at an angle relative to noral then bounces off with the sae speed as it had initially What was the change in oentu of the ball? 2 A tennis ball of ass strikes a wall at an angle relative to noral then bounces off with the sae speed as it had initially What was the change in oentu of the ball in the y direction? A v B 2v C v cos D 2v cos p' p y' p x' p y' p x' p A v B 0 C v D 2v p' p y' p x' p y' p x' p Collisions in Two Diensions Collisions in Two Diensions 20

21 oentu presentation 2016 General Two Diensional Collisions General Two Diensional Collisions Before After p2 = 0 2 p2 =? We'll now consider the ore general case of two objects oving in rando directions in the x y plane and colliding Since there is no absolute reference frae, we'll line up the x axis with the velocity of one of the objects This is not a head on collision note how 1 heads off with a y coponent of velocity after it strikes 2 Also, did you see how we rotated the coordinate syste so the x axis is horizontal? To reduce the aount of variables that ust be specified, we will also assue that ass 2 is at rest The proble now is to find the oentu of 2 after the collision Collisions in Two Diensions Collisions in Two Diensions General Two Diensional Collisions General Two Diensional Collisions Before After Here is the oentu vector breakdown of ass 1 after the collision: 1 2 p2 = p2 =? This will be done by looking at the vectors first oentu ust be conserved in both the x and the y directions Since the oentu in the y direction is zero before the collision, it ust be zero after the collision And, the value that 1 has for oentu in the x direction ust be shared between both objects after the collision and not equally it will depend on the asses and the separation angle 1 2? 2 needs to have a coponent in the y direction to su to zero with 1's final y oentu And it needs a coponent in the x direction to add to 1's final x oentu to equal the initial x oentu of 1: 2 and this is the final oentu for ass 2 by vectorially adding the final p x and p y Collisions in Two Diensions Collisions in Two Diensions 3 After the collision shown below, which of the following is the ost likely oentu vector for the blue ball? General Two Diensional Collisions A B C 1 before 2 after 2? 1 Now that we've seen the vector analysis, let's run through the algebra to find the exact velocity (agnitude and direction) that 2 leaves with after the collision before 200 kg /s after 120 kg /s D E There is a bowling ball with oentu 200 kg /s that strikes a stationary bowling pin and then the bowling ball and pin take off as shown above What is the final velocity of the pin? Collisions in Two Diensions proble Collisions in Two Diensions 21

22 oentu presentation 2016 General Two Diensional Collisions General Two Diensional Collisions before Given: 200 kg /s after 120 kg /s before 200 kg /s after 120 kg /s Use Conservation of Moentu in the x and y directions x direction y direction Find: Collisions in Two Diensions Collisions in Two Diensions before General Two Diensional Collisions 200 kg /s after 120 kg /s Now that the x and y coponents of the oentu of ass 2 have been found, the total final oentu is calculated 4 A 50 kg bowling ball strikes a stationary bowling pin After the collision, the ball and the pin ove in directions as shown and the agnitude of the pin's oentu is 180 kg /s What was the velocity of the ball before the collision? before ball? 531 pin 18 kg /s 30 after Collisions in Two Diensions Collisions in Two Diensions proble Perfect Inelastic Collisions in Two Diensions Perfect Inelastic Collisions in Two Diensions One coon kind of inelastic collision is where two cars collide and stick at an intersection In this situation the two objects are traveling along paths that are perpendicular just prior to the collision Before 1 p 1 2 p After p' Before 2 p 2 1 p 1 p conservation in x: in y: final oentu: final velocity: final direction: 1 2 After p' x p' p' y Inelastic Collisions in Two Diensions Inelastic Collisions in Two Diensions 22

23 oentu presentation Object A with ass 200 kg travels to the east at 100 /s and object B with ass 500 kg travels south at 200 /s They collide and stick together What is the velocity (agnitude and direction) of the objects after the collision? Explosions in Two Diensions The Black object explodes into 3 pieces (blue, red and green) We want to deterine the oentu of the third piece p' 3 p' 2 p' 1 During an explosion, the total oentu is unchanged, since no EXTERNAL force acts on the syste By Newton's Third Law, the forces that occur between the particles within the object will add up to zero, so they don't affect the oentu if the initial oentu is zero, the final oentu is zero the third piece ust have equal and opposite oentu to the su of the other two Inelastic Collisions in Two Diensions proble Explosions in Two Diensions before: p x = p y = 0 p' 2 Explosions in Two Diensions The Black object explodes into 3 pieces (blue, red and green) We want to deterine the oentu of the third piece 6 A stationary cannon ball explodes in three pieces The oenta of two of the pieces is shown below What is the direction of the oentu of the third piece? A p' 1 p' 3 after: p' 1x + p' 2x + p' 3x = 0 p' 1y + p' 2y + p' 3y = 0 In this case the blue and red pieces are oving perpendicularly to each other, so: B C D Explosions in Two Diensions Explosions in Two Diensions proble 7 A stationary 100 kg bob explodes into three pieces A 200 kg piece oves west at 2000 /s Another piece with a ass of 300 kg oves north with a velocity of 1000 /s What is the velocity (speed and direction) of the third piece? Explosions in Two Diensions 23

24 Attachents Eqn1pict