Physics
Elastic Collisions
Elastic collisions are interactions between objects in which both kinetic energy and momentum are conserved. In these collisions, the total kinetic energy of the system before and after the collision remains the same. This means that no energy is lost or transformed into other forms, making elastic collisions distinct from inelastic collisions.
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12 Key excerpts on "Elastic Collisions"
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
When the objects are atoms or subatomic particles, the total kinetic energy of the system is often conserved also. In other words, the total kinetic energy of the particles before the collision equals the total kinetic energy of the particles after the collision, so that kinetic energy gained by one particle is lost by another. In contrast, when two macroscopic objects collide, such as two cars, the total kinetic energy after the collision is generally less than that before the collision. Kinetic energy is lost mainly in two ways: (1) It can be converted into heat because of friction, and (2) it is spent in creating permanent distortion or damage, as in an automobile collision. With very hard objects, such as a solid steel ball and a marble floor, the permanent distortion suffered upon collision is much smaller than with softer objects and, consequently, less kinetic energy is lost. Collisions are often classified according to whether the total kinetic energy changes during the collision: 1. Elastic collision: One in which the total kinetic energy of the system after the col- lision is equal to the total kinetic energy before the collision. 2. Inelastic collision: One in which the total kinetic energy of the system is not the same before and after the collision; if the objects stick together after colliding, the collision is said to be completely inelastic. Problem-Solving Insight As long as the net external force is zero, the conservation of linear momentum applies to any type of collision. This is true whether the collision is elastic or inelastic. 7.3 | Collisions in One Dimension 183 The boxcars coupling together in Figure 7.8 provide an example of a completely inelastic collision. When a collision is completely inelastic, the greatest amount of kinetic energy is lost. Example 7 shows how one particular elastic collision is described using the conserva- tion of linear momentum and the fact that no kinetic energy is lost.
eBook - PDF- John D. Cutnell, Kenneth W. Johnson, David Young, Shane Stadler(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
When the objects are atoms or subatomic particles, the total kinetic energy of the system is often conserved also. In other words, the total kinetic energy of the particles before the collision equals the total kinetic energy of the particles after the collision, so that kinetic energy gained by one particle is lost by another. In contrast, when two macroscopic objects collide, such as two cars, the total kinetic energy after the collision is generally less than that before the collision. Kinetic energy is lost mainly in two ways: (1) It can be converted into heat because of friction, and (2) it is spent in creating permanent distortion or damage, as in an automobile collision. With very hard objects, such as a solid steel ball and a marble floor, the permanent distortion suffered upon collision is much smaller than with softer objects and, consequently, less kinetic energy is lost. Collisions are often classified according to whether the total kinetic energy changes during the collision: 1. Elastic collision: One in which the total kinetic energy of the system after the collision is equal to the total kinetic energy before the collision. 2. Inelastic collision: One in which the total kinetic energy of the system is not the same before and after the collision; if the objects stick together after colliding, the collision is said to be completely inelastic. 7.3 Collisions in One Dimension 185 Problem-Solving Insight As long as the net external force is zero, the conservation of linear momentum applies to any type of collision. This is true whether the collision is elastic or inelastic. The boxcars coupling together in Animated Figure 7.8 provide an example of a completely inelastic collision. When a collision is completely inelastic, the greatest amount of kinetic energy is lost. Example 7 shows how one particular elastic collision is described using the conservation of linear momentum and the fact that no kinetic energy is lost.
eBook - PDF- Paul Peter Urone, Roger Hinrichs(Authors)
- 2012(Publication Date)
- Openstax(Publisher)
• The conservation of momentum principle is valid when considering systems of particles. 8.4 Elastic Collisions in One Dimension • An elastic collision is one that conserves internal kinetic energy. • Conservation of kinetic energy and momentum together allow the final velocities to be calculated in terms of initial velocities and masses in one dimensional two-body collisions. 8.5 InElastic Collisions in One Dimension • An inelastic collision is one in which the internal kinetic energy changes (it is not conserved). • A collision in which the objects stick together is sometimes called perfectly inelastic because it reduces internal kinetic energy more than does any other type of inelastic collision. • Sports science and technologies also use physics concepts such as momentum and rotational motion and vibrations. 8.6 Collisions of Point Masses in Two Dimensions • The approach to two-dimensional collisions is to choose a convenient coordinate system and break the motion into components along perpendicular axes. Choose a coordinate system with the x -axis parallel to the velocity of the incoming particle. • Two-dimensional collisions of point masses where mass 2 is initially at rest conserve momentum along the initial direction of mass 1 (the x -axis), stated by m 1 v 1 = m 1 v′ 1 cos θ 1 + m 2 v′ 2 cos θ 2 and along the direction perpendicular to the initial direction (the y -axis) stated by 0 = m 1 v′ 1y +m 2 v′ 2y . • The internal kinetic before and after the collision of two objects that have equal masses is Chapter 8 | Linear Momentum and Collisions 307 1 2 mv 1 2 = 1 2 mv′ 1 2 + 1 2 mv′ 2 2 + mv′ 1 v′ 2 cos ⎛ ⎝ θ 1 − θ 2 ⎞ ⎠ . • Point masses are structureless particles that cannot spin. 8.7 Introduction to Rocket Propulsion • Newton’s third law of motion states that to every action, there is an equal and opposite reaction. • Acceleration of a rocket is a = v e m Δm Δt − g . • A rocket’s acceleration depends on three main factors.
eBook - PDF- Vern Ostdiek, Donald Bord(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
In this section, we look at collisions from an energy standpoint. In some collisions, the only form of energy involved, before and after, is kinetic energy. In other collisions, forms of energy like potential energy and internal energy play a role. 3.6a Types of Collisions Collisions may be classified as follows. In an elastic collision, kinetic energy is conserved. The total energy is always conserved in both types of collisions, but in Elastic Collisions no energy conver- r r sions take place that make the total kinetic energy after different from the total kinetic energy before. Figure 3.32 illustrates examples of these two types of collisions. Two equal- mass carts traveling with the same speed but in opposite directions collide. In both collisions, the total linear momentum before the collision equals the total after. (This total is equal to zero. Why?) In Figureuni00A03.32a, the carts bounce apart because of a spring attached to one of them. After the collision, each cart has the same speed it had before, but it is going in the opposite direction. Conse- quently, the total kinetic energy of the two carts is the same after the collision as it was before. This is an elastic collision. Figureuni00A03.32b is an example of an inelastic collision. This time the two carts stick together (because of putty on one of them) and stop. The total kinetic An Elastic Collision is one in which the total kinetic energy of the colliding bodies after the collision equals the total s s kinetic energy before the collision. An Inelastic Collision is one in which the total kinetic energy of the colliding bodies after the collision is not equal to the total kinetic energy before. The total kinetic energy after can be greater than, or less than, the total kinetic energy before. DEFINITION Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. WCN 02-300
eBook - PDF- David Halliday, Robert Resnick, Jearl Walker(Authors)
- 2018(Publication Date)
- Wiley(Publisher)
In particular, the velocity v → com of the center of mass can- not be changed by the collision. Elastic Collisions in One Dimension An elastic collision is a special type of collision in which the kinetic energy of a system of colliding bodies is conserved. If the system is closed and isolated, its linear momentum is also conserved. For a one-dimensional col- lision in which body 2 is a target and body 1 is an incoming pro- jectile, conservation of kinetic energy and linear momentum yield the following expressions for the velocities immediately after the collision: v 1f = m 1 − m 2 m 1 + m 2 v 1i (9-67) and v 2f = 2m 1 m 1 + m 2 v 1i . (9-68) Collisions in Two Dimensions If two bodies collide and their motion is not along a single axis (the collision is not head-on), the collision is two-dimensional. If the two-body system is closed and isolated, the law of conservation of momentum applies to the collision and can be written as P → 1i + P → 2i = P → 1f + P → 2f . (9-77) In component form, the law gives two equations that describe the collision (one equation for each of the two dimensions). If the col- lision is also elastic (a special case), the conservation of kinetic energy during the collision gives a third equation: K 1i + K 2i = K 1f + K 2f . (9-78) Variable-Mass Systems In the absence of external forces a rocket accelerates at an instantaneous rate given by Rv rel = Ma (first rocket equation), (9-87) in which M is the rocket’s instantaneous mass (including unexpended fuel), R is the fuel consumption rate, and v rel is the fuel’s exhaust speed relative to the rocket. The term Rv rel is the thrust of the rocket engine. For a rocket with constant R and v rel , whose speed changes from v i to v f when its mass changes from M i to M f , v f − v i = v rel ln M i M f (second rocket equation). (9-88) 245 QUESTIONS Questions 1 Figure 9-23 shows an overhead view of three particles on which external forces act.
eBook - PDF- Jerry B. Marion(Author)
- 2013(Publication Date)
- Academic Press(Publisher)
We shall derive in this chapter those relationships which require only the conservation of momentum and energy, and then we shall examine the features of the collision process which demand that the force law be specified. The discussion here is limited to Elastic Collisions. Of course, collision processes in which kinetic energy is absorbed or rest-mass energy is released are also quite important. However, all of the essential features of two-particle kinematics are adequately demonstrated by Elastic Collisions, and so the more complicated processes are omitted. (The interested reader will find satisfactory treatments in the Suggested References.) We conclude the chapter by discussing briefly some aspects of relativistic collisions. It should be noted that the results which are obtained under the assump-tion only of momentum and energy conservation are valid (in the non-relativistic velocity region) even for quantum mechanical systems, since these conservation theorems are applicable to quantum as well as to classical systems. 11.2 Elastic Collisions—Center-of-Mass and Laboratory Coordinate Systems It has been demonstrated previously on several occasions that the descriptions of many physical processes are considerably simplified if one chooses a coordinate system that is at rest with respect to the center of mass of the system. In the problem which we shall now discuss, viz., the elastic collision of two particles,*! the usual situation (and the one to which we shall confine our attention) is one in which the collision is between a moving particle and a particle at rest. Although it is indeed simpler to describe the effects of the collision in a coordinate system in which the center of mass is at rest, the actual measurements in such a case would be interaction has taken place.
eBook - PDF- Richard C. Hill, Kirstie Plantenberg(Authors)
- 2013(Publication Date)
- SDC Publications(Publisher)
Because the internal impact forces, during the collision, are so large compared to the other external forces acting on the particles, the external forces can be neglected during the collision. 2. Because the duration of the impact is so short, the displacements of the particles during the collision do not need to be considered. 9.4.2) COEFFICIENT OF RESTITUTION Energy is not necessarily conserved during a collision. Energy is often expended in permanently deforming the bodies involved in the collision. Energy can also be expended as heat and sound. A collision in which energy is not conserved is defined as inelastic. A collision in which the two bodies “stick together” following the collision, that is, where the bodies have the same velocity following the collision, is said to be perfectly inelastic (i.e. plastic). Conversely, a collision during which energy is conserved is defined as perfectly elastic. In a perfectly elastic collision, the bodies behave like ideal springs. Work goes into deforming the bodies, but that energy is stored and then released as the bodies rebound. A measure of the elasticity of a collision is its coefficient of restitution given in Equation 9.4-1. The coefficient of restitution relates the incoming and outgoing particle velocities along the line of impact. A perfectly inelastic collision has a coefficient of restitution of e = 0 and a perfectly elastic collision has a coefficient of restitution of e =1. Name examples of impacts that are nearly perfectly elastic and perfectly inelastic.
eBook - PDF- Stephen McKnight, Christos Zahopoulos(Authors)
- 2015(Publication Date)
- Cambridge University Press(Publisher)
1. Perfectly elastic collision : A perfectly elastic collision is one in which momentum and mechanical energy are both conserved in the collision. For example, if we know the masses and initial velocities of two colliding objects in two or three dimensions, we can predict the final velocities as a function of the angle that one of the particles moves after the collision. 2. Perfectly inelastic collision : A perfectly inelastic collision is one in which the bodies collide and stick together. Mechanical energy is not conserved in a perfectly inelastic collision, but we can still predict the final velocity and direction of the combined body if we know the masses and initial velocities of the two bodies before the collision, as in the example below. 23 2.2 The Third Law and momentum conservation Example 2.2 Perfectly inelastic collision Two masses, M 1 with a mass of 10 kg and a velocity of 2 m/s ^ x and M 2 with a mass of 5 kg and a velocity of 6 m/s ^ y , collide. After the collision the two masses stick together as illustrated. What is the velocity (magnitude and direction) of the motion of the combined masses after the collision? M 2 M 1 M 1 M 2 V 1 V 2 V 2 =6 m/s (–y) V 3 V 1 =2 m/s (x) M 1 =10 kg V 1 ^ ^ M 2 =5 kg q Solution: The total x- and y-momentum before the collision is given by: p xi ¼ M 1 V 1 ¼ ð10 kgÞð2 m=sÞ ¼ 20 kg m=s p yi ¼ M 2 V 2 ¼ ð5 kgÞð6 m=sÞ ¼ 30 kg m=s: Since there are no external forces on the system, the momentum doesn’t change, and the momentum after the collision must be the same as the momentum before the collision. The momentum after the collision is just the combined mass times the final velocity. Breaking the final velocity up into its x- and y-components, we get: p xf ¼ ðM 1 þ M 2 Þ V 3 cos θ ¼ 15 V 3 cos θ p yf ¼ ðM 1 þ M 2 Þ ðV 3 sin θÞ ¼ 15 V 3 sin θ: We can set the x-component before the collision equal to the x-component after, and the y-component before equal to the y-component after.
- Raymond Serway, John Jewett(Authors)
- 2018(Publication Date)
- Cengage Learning EMEA(Publisher)
Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 9.4 Collisions in One Dimension 221 Elastic Collisions Consider two particles of masses m 1 and m 2 moving with initial velocities v S 1i and v S 2i along the same straight line as shown in Figure 9.8. The two particles collide head- on and then leave the collision site with different velocities, v S 1f and v S 2f . In an elastic collision, both the momentum and kinetic energy of the system are conserved. Therefore, considering velocities along the horizontal direction in Figure 9.8, we have p i 5 p f S m 1 v 1i 1 m 2 v 2i 5 m 1 v 1f 1 m 2 v 2f (9.16) K i 5 K f S 1 2 m 1 v 1i 2 1 1 2 m 2 v 2i 2 5 1 2 m 1 v 1f 2 1 1 2 m 2 v 2f 2 (9.17) Because all velocities in Figure 9.8 are either to the left or the right, they can be 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- tive if it moves to the left. In a typical problem involving Elastic Collisions, there are two unknown quan- tities, and Equations 9.16 and 9.17 can be solved simultaneously to find them. An alternative approach, however—one that involves a little mathematical manipula- tion of Equation 9.17—often simplifies this process.
eBook - PDF- Raymond Serway, Chris Vuille(Authors)
- 2017(Publication Date)
- Cengage Learning EMEA(Publisher)
Conservation of momentum can be written mathematically for this case as m 1 v S 1i 1 m 2 v S 2i 5 m 1 v S 1f 1 m 2 v S 2f 2f 2 [6.7] collision, the colliding objects stick together. In an elastic collision, both the momentum and the kinetic energy of the system are conserved. A one - dimensional elastic collision between two objects can be solved by using the conservation of momentum and conservation of energy equations: m 1 v 1i 1 m 2 v 2i 5 m 1 v 1f 1 m 2 v 2f 2f 2 [6.10] 1 2 m 1 v 1i 2 1 1 2 m 2 v 2i 2 5 1 2 m 1 v 1f 2 1 1 2 m 2 v 2f 2f 2 2 [6.11] The following equation, derived from Equations 6.10 and 6.11, is usually more convenient to use than the original conservation of energy equation: v 1i 2 v 2i 5 2(v 1f 2 v 2f 2f 2 ) [6.15] These equations can be solved simultaneously for the unknown velocities. Energy is not conserved in inelastic col- lisions, so such problems must be solved with Equation 6.10 alone. 6.4 Glancing Collisions In glancing collisions, conservation of momentum can be applied along two perpendicular directions: an x - axis and a y - axis. Problems can be solved by using the x - and y - components of Equation 6.7. Elastic two - dimensional col- lisions will usually require Equation 6.11 as well. (Equation 6.15 doesn’t apply to two dimensions.) Generally, one of the two objects is taken to be traveling along the x - axis, under- r- r going a deflection at some angle u after the collision. The final velocities and angles can be found with elementary trigonometry. 6.5 Rocket Propulsion From conservation of momentum, the change in a rocket’s velocity as it ejects fuel is given by v f v f v 2 v i 5 v e ln a M i i M f M f M b b [6.20] where M i i and and M f M f M are the rocket’s initial and final masses, are the rocket’s initial and final masses, respectively, and v e e is the fuel’s exhaust speed relative to the rocket.
- Robert Ehrlich(Author)
- 2020(Publication Date)
- Princeton University Press(Publisher)
In addition to observing one ball colliding with a target ball of equal mass, you should also observe the following: (1) Elastic collision between two balls of equal mass having equal and opposite velocities. (2) Elastic collision between one ball and a row of two or more stationary balls of equal mass. (3) Elastic collision between a (light) Ping-Pong ball and a (heavy) steel ball moving toward each other with equal speeds. You should find that the Ping-Pong ball rebounds with a much increased speed (ideally, three times its orig-inal speed if spin is ignored), and the steel ball's speed is essentially unchanged by the collision. (4) Elastic collision between two metal balls whose mass ratio is 3 : 1, moving toward each other with equal speeds. If momentum and energy are both conserved, and spin is 59 Energy and Momentum Conservation ignored, the heavy ball should be at rest after the collision, and the light ball should double its speed. In practice, the spin of the heavy ball results in its possessing some for-ward motion after the collision. (5) Inelastic collision between two balls of equal mass with equal and opposite velocities. For the inelastic collision you can use two no-bounce balls (see A.I). Ping-Pong balls sawed in half, filled with clay, and glued back to-gether are also good. When observing collisions between balls of very different mass, it is best to use balls of roughly the same size and different densities, rather than different sizes and the same density, so the balls collide nearly center to center. E.7. Momentum conservation using an embroidery hoop Demonstration When two touching, stationary balls inside a horizontal em-broidery hoop are driven apart with a sharp vertical blow, momentum conservation requires that the ratio of the angles they travel before colliding equal the inverse ratio of their masses.
eBook - PDFDynamics of Particles and the Electromagnetic Field
(With CD-ROM)
- Slobodan Danko Bosanac(Author)
- 2005(Publication Date)
- WSPC(Publisher)
Chapter 14 Inelastic Scattering 14.1 Laboratory Coordinate System Scattering by the center of force is only a model that describes an encounter of two particles, in which one is assumed to be infinitely massive. As the result there is no energy exchange between them, and in the real circum- stances this effect needs to be taken into account. The elastic collision, as the former is called, is only used as a mathematical convenience. Position of particles is normally specified with respect to some fixed reference frame, called laboratory reference frame, which is defined inde- pendently of their whereabouts. In the laboratory reference frame particle 1 has the coordinate F1 and the particle 2 has the coordinate F2, whilst the respective velocities are and i&. Typical of the collisions in this reference frame is that the kinetic energy of one particle is not the same before and after the encounter with the other, which is called the inelastic collision (scattering). The force between the two particles is a function of the dif- ference r' = F'l - F'2, and therefore it would be natural to use it as a new coordinate. For a complete transformation one needs another conveniently defined coordinate so that the equations of motion separate into two in- dependent sets, where in one the force between the particles appears and in the other it does not. From this requirement the second coordinate is defined uniquely and it is given by in which case the equations of motion are 307 308 Dynamics of Particles and the Electromagnetic Field This coordinate is that of the center of mass of the two particles, and the new reference frame is called the center of mass coordinate system. Dynam- ics in the coordinate r‘ is identical to the dynamics of a single particle that is scattered by a fixed center of force, but with the reduced mass p = v, where M = ml +m2 is the total mass of the system.
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