Momentum. February 15, Table of Contents. Momentum Defined. Momentum Defined. p =mv. SI Unit for Momentum. Momentum is a Vector Quantity.

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1 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 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: Moentu Defined 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. Moentu Defined Define a new quantity, oentu (p), that takes these observations into account: oentu = ass velocity p =v What's wrong with this equation? 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 SI Unit for Moentu There is no specially naed unit for oentu - so there is an opportunity for it to be naed a renowned physicist! We use the product of the units of ass and velocity. ass x velocity kg /s Since oentu is a vector, you need to specify a direction - such as to the right, up, down, to the east, or use positive (to the right) or negative (to the left), or any direction that is relevant to the proble.

2 1 Which object or objects have the greatest oentu? A A large trailer truck oving at 30 /s. B A otorcycle oving at 30 /s. C The Epire State Building. D Choices A and B have the sae oentu. Ipulse-Moentu Equation Return to Table of Contents Change in Moentu Moentu Change = Ipulse Suppose that there is an event that changes an object's oentu. Moentu change equation: fro - the initial oentu (just the event) by - the change in oentu to - the final oentu (just the event) The equation for oentu change is: 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 ter? is related to the external force. Moentu Change = Ipulse Let's now use Newton's Second Law, and see if we can relate the external force in an explicit anner: to 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: 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.

3 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. 2 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. A Increase the applied force. B Increase the tie that the force is applied. C Maintain the sae applied force. D Decrease the tie that the force is applied. 3 Fro which law or principle is the Ipulse-Moentu equation derived fro? 4 Can the ipulse applied to an object be negative? Why or why not? Give an exaple to explain your answer. A Conservation of Energy. B Newton's First Law. C Newton's Second Law. D Conservation of Moentu. Effect of Collision Tie on Force Ipulse = 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! Let's just work with the agnitudes of the ipulse and oentu, so the vector sign will be dropped. Every Day Applications Ipulse = 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 parachuting artial arts hitting a baseball catching a baseball

4 Every Day Applications Car Air Bags Let's analyze two specific cases fro the previous list: car air bags hitting a baseball Whenever you have an equation with three variables, you have to decide which value 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. 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? For the bat hitting a ball, we'll fix F, vary Δt and see the ipact on Δp. Car Air Bags Car Air Bags 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 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? 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 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. 5 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. 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.

5 6 Which of the following variables are fixed, using the Ipulse - Moentu equation, when analyzing a oving car that strikes a barrier and coes to rest? A Force delivered to the passengers. B Interval of tie the passengers coe to rest. C Moentu change of the car. D Acceleration of the passengers within the car. Hitting a Baseball 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. What is the goal of the batter? Hitting a Baseball 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. Hitting a Baseball 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. 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). Every Day Applications 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. 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 parachuting artial arts catching a basebal 7 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 change in oentu of a collision. D decrease the ipulse in a collision. B

6 8 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. A change in oentu B force on the passengers C ipact tie on the passengers D final oentu 9 In order to increase the final oentu of a golf ball, the golfer should: Select two answers: A aintain the speed of the golf club the ipact (follow through). B Hit the ball with a greater force. C Decrease the tie of contact between the club and the ball. D Decrease the initial oentu of the golf club. Graphical Interpretation of Ipulse Graphs are powerful tools to help solve physics probles. Graphical Interpretation of Ipulse F [N] 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. F Area = 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. Ipulse (Change in Moentu) t t [s] Start with representing a constant force over a tie interval, Δt, and plot Force vs. tie on a graph. Note, the area under the Force-Tie graph is: height base = F t = I (ipulse) = p (change in oentu) Graphical Interpretation of Ipulse 10 Using the F-t graph shown, what is the change in oentu during the tie interval fro 6 to 10 s? F [N] F Area = Ipulse (Change in Moentu) t t [s] 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.

7 The Moentu of a Syste of Objects 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. Return to Table of Contents It's critically iportant to note that oenta add as vectors, not as scalars. The Moentu of a Syste of Objects In order to deterine the total oentu of a syste: Deterine a direction to be considered positive. Assign positive values to oenta in that direction. Assign negative values to oenta in the opposite direction. + Add the oenta to get the total oentu of the syste. - Exaple Deterine the oentu of a syste of two objects:, has a ass of 6 kg and a velocity of 13 /s towards the east and, has a ass of 14 kg and a velocity of 7 /s towards the west. 6 kg 14 kg 13 /s 7 /s East (+) = 6 kg v1 = 13 /s = 14 kg v2 = 7 /s Conservation Laws Conservation of Moentu Deo Soe of the ost powerful concepts in science are called "conservation laws". This was covered in the Work and Energy unit - please refer back to that for ore detail. Here is a suary. Conservation laws: apply to closed systes - where the objects only interact with each other and nothing else. enable us to solve probles without worrying about the details of an event. Return to Table of Contents

8 Moentu is Conserved In the last unit we learned that energy is conserved. Like energy, oentu is a conserved property of nature. This eans: Moentu is not created or destroyed. Conservation of Moentu Conservation of Moentu is used to explain and predict the otion of a syste of objects. As with energy, it will only be necessary to copare the syste at two ties: just and just an event. The total oentu in a closed syste is always the sae. The only way the oentu of a syste can change is if oentu is added or taken away by an outside force. Here are two spheres colliding with each other and rebounding. If there are no external forces (this is a closed syste), then the oentu and the collision is the sae. Conservation of Moentu Conservation of Moentu The oentu of an object changes when it experiences an ipulse (I=FΔt): If there is no net external force on the syste, then F = 0, and since I = FΔt, I is also zero: This ipulse arises when a non-zero external force acts on the object. This is exactly the sae for a syste of objects. Since psyste, 0 = psyste, f, the oentu of the syste is conserved, that is, it does not change. 11 Why don't the internal forces of a syste change the oentu of the syste? 12 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. C The oentu of the syste decreases. D The oentu of the syste increases.

9 Types of Collisions Objects in an isolated syste can interact with each other in two basic ways: Types of Collisions 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. Return to Table of Contents Conservation of oentu and change in kinetic energy predict what will happen in these events. Elastic Collisions 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. 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? Explosions In both an inelastic collision and an explosion, energy changes between kinetic and 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. 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

10 13 In the absence of external forces, oentu is conserved in which of the following situations? A Elastic collisions only. B Inelastic collisions only. C Explosions only. D Elastic and Inelastic collisions and explosions. Collisions in Two Diensions Return to Table of Contents Conservation of Moentu in Two Diensions General Two Diensional Collisions 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 = can be solved independently for each coponent: This, of course also applies to three diensions, but we'll stick with two for this chapter! 1 2 Consider a syste of two objects oving in rando directions in a two diensional plane and colliding. Assue there are no external forces acting on the syste. Since there is no absolute reference frae, we'll line up the x- axis with the velocity of one of the objects. General Two Diensional Collisions General Two Diensional Collisions Before After Before After p2 = 0 2 =? 1 2 p2 = =? This is not a head on collision - note how heads off with a y coponent of velocity it strikes. Also, did you see how we rotated the coordinate syste so the x axis is horizontal? To work a sipler proble, we'll assue that ass 2 is at rest. Find the oentu of the collision. Look at the vectors first - oentu ust be conserved in both the x and the y directions. Since the syste oentu in the y direction is zero the collision, it ust be zero the collision. The value that has for oentu in the x direction ust be shared between both objects the collision - and not equally - it will depend on the asses and the separation angle.

11 General Two Diensional Collisions Here is the oentu vector breakdown of ass 1 the collision:? needs to have a coponent in the y direction to su to zero with 's final y oentu. And it needs a coponent in the x direction to add to 's final x oentu to equal the initial x oentu of : and this is the final oentu for ass 2 by vectorially adding the final px and py. 14 After the collision shown below, which of the following is the ost likely oentu vector for the blue ball? A B C D E? General Two Diensional Collisions General Two Diensional Collisions Now that we've seen the vector analysis, let's run through the algebra to find the oentu (agnitude and direction) that 2 leaves with the collision kg-/s 12.0 kg-/s ball 60.0 Given: 20.0 kg-/s 12.0 kg-/s 60.0 There is a bowling ball ( 1 ) with oentu 20.0 kg-/s that strikes a stationary bowling pin ( 2 ) and then the bowling ball and pin take off as shown above. What is the final oentu of the pin? pin Find: General Two Diensional Collisions General Two Diensional Collisions 20.0 kg-/s 12.0 kg-/s kg-/s 12.0 kg-/s 60.0 Use Conservation of Moentu in the x and y directions. x direction y-direction Now that the x and y coponents of the oentu of ass 2 have been found, the final oentu of the bowling pin is calculated.

12 Two Diensional Collisions Proble Solution 15 A 5.0 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 18 kg-/s. What was the velocity of the ball the collision?? ball 53.1 pin 30.0 pin =18 kg-/s Start with the given quantities - and note that this proble will be worked backwards; the final, not the initial conditions are specified.? ball 53.1 pin 30.0 Given: Find: pin =18 kg-/s Two Diensional Collisions Proble Solution? ball 53.1 pin 30.0 pin =18 kg-/s Two Diensional Collisions Proble Solution? ball 53.1 pin 30.0 Find 1x and 2x pin =18 kg-/s Conservation of Moentu in x and y directions: To find the initial velocity of the bowling ball, we need to find p1x, and then divide it by the ass of the ball. We now have to find 1y in order to find 1x Two Diensional Collisions Proble Solution? ball 53.1 pin 30.0 pin =18 kg-/s Perfect Inelastic Collisions in Two Diensions A coon perfect inelastic collision is where two cars collide and stick at an intersection. The cars are traveling along paths that are perpendicular just prior to the collision. Before p2 After p1 Final answer!

13 Perfect Inelastic Collisions in Two Diensions Before p1 p2 After x y 16 Object A with ass 20.0 kg travels to the east at 10.0 /s and object B with ass 5.00 kg travels south at 20.0 /s. They collide and stick together. What is the velocity (agnitude and direction) of the objects the collision? p-conservation in x: in y: final oentu: final velocity: final direction: Explosions in Two Diensions 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, given the oentu of the pieces 1 and 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. 3 1 Can you see why the third piece oves the way it is shown? 3 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 pieces. : px = py = Explosions in Two Diensions 1 : 1x + 2x + 3x = 0 1y + 2y + 3y = 0 The Black object explodes into 3 pieces (blue, red and green). We want to deterine the oentu of the third piece. In this case the blue and red pieces are oving perpendicularly to each other, so: 17 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 B C D

14 18 A stationary 10.0 kg bob explodes into three pieces. A 2.00 kg piece oves west at /s. Another piece with a ass of 3.00 kg oves north with a velocity of /s. What is the velocity (speed and direction) of the third piece? Exaple: Collision with a Wall A golf ball collides elastically with a rigid wall that does not ove (ignore any ipact on the internal energy of the wall). The angle of incidence equals the rebound angle of the ball. Given the constraints, what can be said about the oenta in the x and y directions? py' px' py px The solid lines represent the oentu of the ball (blue - prior to collision, red - the collision). The dashed lines are the x and y coponents of the oentu vectors. p Exaple: Collision with a Wall Exaple: Collision with a Wall An external force fro the wall is being applied to the ball in order to reverse its direction in the x axis. Since we're assuing an elastic collision, the ball bounces off the wall with the sae speed that it struck the wall with. Hence, the agnitude of the initial oentu and the final oentu is equal: py' px' px py p Apply the Ipulse Moentu Theore in two diensions, where the wall is exerting the external force: py' px' px py p Now it's tie to resolve each oentu into coponents along the x and y axis. What does this tell us? Exaple: Collision with a Wall Moentu is conserved in the y direction, py but it is not conserved in the x direction - the only force is the force exerted by the wall in the -x direction - the Noral Force. The initial and final oentu in the y direction are the sae and subtract out. The initial and final oentu in the x direction are equal in agnitude, but opposite in direction - and add up to a value in the -x direction. py' px' px p 19 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 is the change in oentu of the ball in the x direction? A 0 B -2v C -v cos D -2v cos py' px' p y p x p

15 20 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 is the change in oentu of the ball in the y direction? A 0 B -2v C -v cos D -2v cos py' px' p y p x p