1、Chapter 2 Motion in one dimensionKinematicsDynamicsAt any particular time t,the particle can be located by its x,y and z coordinates,whichare the three components of the position vector :where ,and are the cartesian unit vectors.Section2-3 Position,velocity and acceleration vectors1.Position vectork
2、jir+=kzjyixrxyzFig 2-11ijkrOWe defined the displacement vector as the change in position vector from t1to t2.2.Displacement(位移位移)12rrr=rNote:1)Displacement is not the same as the distancetraveled by the particle.2)The displacement is determined only by the starting and ending points of the interval.
3、yz1r1t2tt=t=Fig 2-12x2rsOr222:Magnitudezyxr+=Direction:from start point to end point kzjyixrif1111+=kzjyixr2222+=12rrr=Then the displacement is kzzjyyixx)()()(121212+=The relationship between and :rs1r1P2r2PrxyOzsrs In general,Can?sr=Yes,for two cases:1)1D motion without changing direction2)When aft
4、er take limit:,0tdsrd=The difference between and ():rrrr1r1P2r2PrxyOrz212121zyx+222222zyx+=rNote:magnitude of rr:the change of lengthof position vectorsr)(rrrWhen after take limit:,0trddrrdr|rdrdss drr rdr|rd=3.velocity and speeda.The average velocity in any interval is defined to be displacement di
5、vided by the time interval,(2-7)when we use the term velocity,we mean the instantaneous velocity.b.To find the instantaneous velocity,we reduce the size of the time interval ,that isand then .0t0rdtrdtrttrttrtvtt=+=limlim00)()()(trvav=t(2-9)+=kdtdzjdtdyidtdxdtrdvThe vector can also be written in ter
6、ms of its components as:(2-11)vkvjvivvzyx+=dtdxvx=dtdyvy=dtdzvz=(2-12)+=kdzjdyidxrdIn cartesian coordinates:DiscussionThe position vector of a moving particle at a moment is .The magnitude of the velocity of the particle at the moment is:),(yxrtrddtrd|d|（A）（B）（B）（B）trdd22)dd()dd(tytx+（C）（D）c.The ter
7、ms average speed(平均速率)and speed（速率）:Average speed:tsvav=Thus,|trvvavav=is the total distance traveled.sdtdsv=Speed:|vv=|vdtrd=d.AccelerationWe define the average acceleration as the change in velocity per unit time,or(2-14)And instantaneous acceleration(2-16)By analogy with Eq(2-12),we can write the
8、 components acceleration vector as(2-17)tvaav=dtvdtvat=lim0dtdvaxx=dtdvayy=dtdvazz=Sample problem 2-4A particle moves in the x-y plane and ,where ,and .Find the position,velocity,and acceleration of the particle when t=3s.BtAttx+=3)(DCtty+=2)(3/00.1smA=smB/0.32=mD0.12=3/0.5smC=+=+=jCtiBAtjvivvyx2)3(
9、2+=+=jCiAtjaiaayx26+=+=jDCtiBtAtjyixr)()(23Solution:=+=jmimrst)57()69(|3=+=jsmismvst)/30()/5(|3=+=jsmismast)/10()/18(|223Sample problemHow do the velocity andthe acceleration vary withtime if position x(t)is known?x(t)dtdxvx=dtdvaxx=Colonel J.P.Stapp was in his braking rocket sledCan you tell the di
10、rection of the acceleration from the figures?His body is an accelerometer not a speedometer.OutinSection 2-4 One-dimensional kinematicsIn one-dimensional kinematics,a particle can move only along a straight line.We can describe the motion of a particle in two ways:with mathematical equationsand with
11、 graphs.ttx0(a)(b)BA0vxFig 2-151.Motion at constant velocitySuppose a puck(冰球)moves along a straight line,which we will use as the x-axis.cvx=cdtdxvx=vtxx+=0Two examples of accelerated motion are(2-20)（2-21）2)(CtBtAtx+=)cos()(tDtx=2.Accelerated motion(变速运动)2-5 Motion with constant accelerationdtdvax
12、x=dtadvxx=dtadtadvxxx0 xxvtav+=Note:the initial velocity must be known in the calculation.0vor(2-26)Can we obtain and x(t)from?xa)(tvxxaLets assume our motion is along the x axis,andrepresents the x component of the acceleration.It is the similar way to find x(t)from v(t).dtdxvx=dtvdxx=20021)(ctatvd
13、ttavdxxx+=+=22100tatvxxx+=orNote:the initial position x0and velocity must be known in the calculation.0v)(ta)(trDerivativeIntegral()()tvRelationship between ,and)(ta)(tr)(tvDerivative)0(rIntegral()0(vDiscussiontaddv=+=ttta0d)(0v(t)vtddrv=+=ttrtr00d)(t)vOne example:动画库力学夹1-02质点运动的描述2.exe 4(例3)2-6 Fre
14、ely falling bodies Aristotle(384-322 B.C.)thought the heavier objectswould fall more rapidly because of their weight.Galileo(1564-1642)made correct assertion,that in the absence of air resistance all objects fall with the same speed.In 1971,astronaut David Scott dropped a featherand a hammer on the
15、airless Moon,and he observed that they reached the surface at about the same time.Feather and hammer experiment carried on the airless moonFeather and apple exp.conducted in vacuum lab.Niagara FallIf the height of the fall is 48 m,how long will the person reach the fall bottom and what is his final
16、speed?Is it dangerousto free fall from its top?!Chapter 3 Force and Newtons lawsIssac Newton(16421727）Galileo Galilei (15641642）Section 3-1 Classical mechanics The approach to the dynamics we consider here is generally called classical mechanics.In this chapter,we will study in detail the bases of classical mechanics:Newtons three laws.Classical mechanics was found not to describe well the motions in certain realms.For ordinary objects,classical mechanics is important and very useful.Section 3-2
