1、Lecture7Lecture7Linear Momentum & Linear Momentum & CollisionsCollisions第1頁(yè)/共38頁(yè)Key words (linear) momentum 動(dòng)量 impulse 沖量 impulse-momentum theorem 動(dòng)量定理 time-average force 平均沖力 interaction 相互作用 air bags 氣囊 particles system: composed of some particles 質(zhì)點(diǎn)系 internal forces: among particles 內(nèi)力 ex

2、ternal forces: acted on from surroundings 外力第2頁(yè)/共38頁(yè)Momentum From Newtons laws: force must be present to change an objects velocity (speed and/or direction) Wish to consider effects of collisions and corresponding change in velocity Method to describe is to use concept of linear momentumscalarvector

3、Golf ball initially at rest, so some of the KE of club transferred to provide motion of golf ball and its change in velocity第3頁(yè)/共38頁(yè)MomentumlVector quantity, the direction of the momentum is the same as the velocityslApplies to two-dimensional motion as wellSize of momentum: depends upon mass depend

4、s upon velocity 第4頁(yè)/共38頁(yè)ImpulselIn order to change the momentum of an object (say, golf ball), a force must be appliedlThe time rate of change of momentum of an object is equal to the net force acting on itpGives an alternative statement of Newtons second lawp(F*t) is defined as the impulsepImpulse

5、is a vector quantity, the direction is the same as the direction of the force第5頁(yè)/共38頁(yè)ConcepTestSuppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?1. It takes less t

6、ime to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.第6頁(yè)/共38頁(yè)ConcepTestSuppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop t

7、hem compare?1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball. Note: Because force equals the time rate of change of momentum, the two balls loose momentum at the same rate. If both balls initially had the same momenta, it ta

8、kes the same amount of time to stop them.第7頁(yè)/共38頁(yè)P(yáng)roblem: Teeing OffA 50-g golf ball at rest is hit by “Big Bertha” club with 500-g mass. After the collision, golf leaves with velocity of 50 m/s.a) Find impulse imparted to ballb) Assuming club in contact with ball for 0.5 ms, find average force acti

9、ng on golf ball.第8頁(yè)/共38頁(yè)P(yáng)roblem: teeing offGiven:mass: m=50 g = 0.050 kgvelocity: v=50 m/sFind: impulse=?Faverage=?1. Use impulse-momentum relation:2. Having found impulse, find the average force from the definition of impulse:smkgsmkgmvmvpimpulseif50. 2050050. 0NssmkgtpFthustFp331000. 5105 . 050. 2

10、, Note: according to Newtons 3rd law, that is also a reaction force to club hitting the ball:iiffififRVMvmVMvmorVMVMvmvmortFtF,of clubCONSERVATION OF MOMENTUM第9頁(yè)/共38頁(yè)Conservation of Momentum Definition: an isolated system is the one that has no external forces acting on it A collision may be the res

11、ult of physical contact between two objects “Contact” may also arise from the electrostatic interactions of the electrons in the surface atoms of the bodies第10頁(yè)/共38頁(yè)Conservation of Momentum The principle of conservation of momentum states when no external forces act on a system consisting of two obj

12、ects that collide with each other, the total momentum of the system before the collision is equal to the total momentum of the system after the collision第11頁(yè)/共38頁(yè)Conservation of Momentum Mathematically: Momentum is conserved for the system of objects The system includes all the objects interacting w

13、ith each other Assumes only internal forces are acting during the collision Can be generalized to any number of objects第12頁(yè)/共38頁(yè)P(yáng)roblem: Teeing Off (cont.)Lets go back to our golf ball and club problem:factor of 10 times smaller第13頁(yè)/共38頁(yè)ConcepTestSuppose a person jumps on the surface of Earth. The E

14、arth1. will not move at all2. will recoil in the opposite direction with tiny velocity3. might recoil, but there is not enough information provided to see if that could happened第14頁(yè)/共38頁(yè)ConcepTestSuppose a person jumps on the surface of Earth. The Earth1. will not move at all2. will recoil in the op

15、posite direction with tiny velocity3. might recoil, but there is not enough information provided to see if that could happened Note: momentum is conserved. Lets estimate Earths velocity after a jump by a 80-kg person. Suppose that initial speed of the jump is 4 m/s, then:tiny negligible velocity, in

16、 opposite direction第15頁(yè)/共38頁(yè)Types of Collisions Momentum is conserved in any collisionwhat about kinetic energy? Inelastic collisions Kinetic energy is not conserved Some of the kinetic energy is converted into other types of energy such as heat, sound, work to permanently deform an object Perfectly

17、 inelastic collisions occur when the objects stick together Not all of the KE is necessarily lost第16頁(yè)/共38頁(yè)P(yáng)erfectly Inelastic Collisions:When two objects stick together after the collision, they have undergone a perfectly inelastic collisionSuppose, for example, v2i=0. Conservation of momentum becom

18、es第17頁(yè)/共38頁(yè)P(yáng)erfectly Inelastic Collisions:What amount of KE lost during collision?lost in heat/”gluing”/sound/第18頁(yè)/共38頁(yè)More Types of Collisions Elastic collisions both momentum and kinetic energy are conserved Actual collisions Most collisions fall between elastic and perfectly inelastic collisions第

19、19頁(yè)/共38頁(yè)More About Elastic Collisions Both momentum and kinetic energy are conserved Typically have two unknowns Solve the equations simultaneously第20頁(yè)/共38頁(yè)P(yáng)roblem Solving for One -Dimensional Collisions Set up a coordinate axis and define the velocities with respect to this axis It is convenient to

20、 make your axis coincide with one of the initial velocities In your sketch, draw all the velocity vectors with labels including all the given information第21頁(yè)/共38頁(yè)Sketches for Collision ProblemsDraw “before” and “after” sketchesLabel each object include the direction of velocitykeep track of subscrip

21、ts第22頁(yè)/共38頁(yè)Sketches for Perfectly Inelastic CollisionsThe objects stick togetherInclude all the velocity directionsThe “after” collision combines the masses第23頁(yè)/共38頁(yè)P(yáng)roblem Solving for One-Dimensional Collisions, cont. Write the expressions for the momentum of each object before and after the collis

22、ion Remember to include the appropriate signs Write an expression for the total momentum before and after the collision Remember the momentum of the system is what is conserved第24頁(yè)/共38頁(yè)P(yáng)roblem Solving for One-Dimensional Collisions, final If the collision is inelastic, solve the momentum equation fo

23、r the unknown Remember, KE is not conserved If the collision is elastic, you can use the KE equation to solve for two unknowns第25頁(yè)/共38頁(yè)Glancing Collisions For a general collision of two objects in three-dimensional space, the conservation of momentum principle implies that the total momentum of the

24、system in each direction is conserved Use subscripts for identifying the object, initial and final, and components第26頁(yè)/共38頁(yè)Glancing CollisionsThe “after” velocities have x and y componentsMomentum is conserved in the x direction and in the y directionApply separately to each direction第27頁(yè)/共38頁(yè)P(yáng)roble

25、m Solving for Two-Dimensional Collisions Set up coordinate axes and define your velocities with respect to these axes It is convenient to choose the x axis to coincide with one of the initial velocities In your sketch, draw and label all the velocities and include all the given information第28頁(yè)/共38頁(yè)P(yáng)

26、roblem Solving for Two-Dimensional Collisions, cont Write expressions for the x and y components of the momentum of each object before and after the collision Write expressions for the total momentum before and after the collision in the x-direction Repeat for the y-direction第29頁(yè)/共38頁(yè)P(yáng)roblem Solving

27、 for Two-Dimensional Collisions, final Solve for the unknown quantities If the collision is inelastic, additional information is probably required If the collision is perfectly inelastic, the final velocities of the two objects is the same If the collision is elastic, use the KE equations to help so

28、lve for the unknowns第30頁(yè)/共38頁(yè)Rocket propulsion The operation of a rocket depends upon the law of conservation of linear momentum as applied to a system of particles, where the system is the rocket plus its ejected fuel. Because the gases are given momentum when they are ejected out of the engine, th

29、e rocket receives a compensating momentum in the opposite direction. Therefore, the rocket is accelerated as a result of the “push,” or thrust, from the exhaust gases.第31頁(yè)/共38頁(yè)Rocket Propulsion The rocket is accelerated as a result of the thrust of the exhaust gases This represents the inverse of an

30、 inelastic collision Momentum is conserved Kinetic Energy is increased (at the expense of the stored energy of the rocket fuel)第32頁(yè)/共38頁(yè)Rocket Propulsion The operation of a rocket depends on the law of conservation of momentum as applied to a system, where the system is the rocket plus its ejected fuel This is different than