EDWARD CARES v
Particlev Kinematics
3.1 Particlev Kinematics
vParticlesvmayvbe vdescribed vasvpoint-
likevobjectsvwithvmassvbutvnegligiblevsizevandvinternalvstructure.vKinematicsvisvthevmathematicalvdescriptionv
ofvanvobject'svmotionvwithoutvconsideringvthevcausevofvthevmotion.vThisvsectionvcoversvone-
dimensionalvstraight-linevmotionvandvitsvgeneralizationvtovtwovandvthreevdimensions.
3.1.1 One-Dimensionalv Rectilinearv Motion
Rectilinearvmotionvisvthevmotionvofvavparticlevalongvavstraight vlinev(e.g.,vavballvrollingvonvthevfloor,vavcarvonvav
straightvroad,vorvanvobjectvinvfreevfall).vRectilinearvmotionvdescribesvthevmovementvofvavparticlevalongvavstraightv
line.vThisvservesvasvthevfoundationalvmodelvforvunderstandingvmorevcomplexvmotions.vExamplesvinclude vavcar
vmoving vonvavstraight vroad, vavtrainvonvavstraight vtrack, vor van vobject vfalling vverticallyvundervgravity.
KeyvDefinitions:
Displacement v(∆𝑥):vAvvectorvrepresentingvthevchangevinvposition,v∆𝑥v=v𝑥𝑓v−v𝑥𝑖.
Distance:vThevtotalvpathvlengthvcoveredv(avscalarvquantity).
Velocityv(𝑣):vThevratevofvchangevofvdisplacementvwithvtimev(𝑣v=v∆𝑥).
∆𝑡
Speed:vThevmagnitudevofvvelocityv(scalar).
Accelerationv(𝑎):vThevratevofvchangevofvvelocityvwithvtimev(𝑎v=v∆𝑣).
∆𝑡
Figurev1:Displacement-
TimevGraphvforvKeyvDifference:
Distancevisvalwaysvpositivevorvzero,vwhilevdisplacementvcanvbevpositive,vnegative,vorvzero.vAvroundvtripvyields
vzero vdisplacement vbut va vpositive vdistance.
Uniformv Acceleration:
Whenvaccelerationvisvconstant,vthevfollowingvequationsvofvmotionvapply:
1. 𝑣v =v𝑢v+v𝑎𝑡
2. 𝑠v=v𝑢𝑡v+v1v𝑎𝑡2
2
3. 𝑣2v =v 𝑢2v +v2𝑎𝑠
(𝑢+𝑣)v
4. 𝑠v=v 𝑡
2
Where:
𝑢v=vinitialvvelocity
𝑣v=vfinalvvelocity
𝑎v=vacceleration
𝑠v =vdisplacement
𝑡v =vtime
Figurev 2:Velocity-Timev Graphv forv Uniformv Acceleration
Examplev1:vAvcarvstartsvfromvrestvandvtravelsv0.25vkmvinv25vsvatvconstantvacceleration. vFind:
1
, a) Thevacceleration
2
Particlev Kinematics
3.1 Particlev Kinematics
vParticlesvmayvbe vdescribed vasvpoint-
likevobjectsvwithvmassvbutvnegligiblevsizevandvinternalvstructure.vKinematicsvisvthevmathematicalvdescriptionv
ofvanvobject'svmotionvwithoutvconsideringvthevcausevofvthevmotion.vThisvsectionvcoversvone-
dimensionalvstraight-linevmotionvandvitsvgeneralizationvtovtwovandvthreevdimensions.
3.1.1 One-Dimensionalv Rectilinearv Motion
Rectilinearvmotionvisvthevmotionvofvavparticlevalongvavstraight vlinev(e.g.,vavballvrollingvonvthevfloor,vavcarvonvav
straightvroad,vorvanvobjectvinvfreevfall).vRectilinearvmotionvdescribesvthevmovementvofvavparticlevalongvavstraightv
line.vThisvservesvasvthevfoundationalvmodelvforvunderstandingvmorevcomplexvmotions.vExamplesvinclude vavcar
vmoving vonvavstraight vroad, vavtrainvonvavstraight vtrack, vor van vobject vfalling vverticallyvundervgravity.
KeyvDefinitions:
Displacement v(∆𝑥):vAvvectorvrepresentingvthevchangevinvposition,v∆𝑥v=v𝑥𝑓v−v𝑥𝑖.
Distance:vThevtotalvpathvlengthvcoveredv(avscalarvquantity).
Velocityv(𝑣):vThevratevofvchangevofvdisplacementvwithvtimev(𝑣v=v∆𝑥).
∆𝑡
Speed:vThevmagnitudevofvvelocityv(scalar).
Accelerationv(𝑎):vThevratevofvchangevofvvelocityvwithvtimev(𝑎v=v∆𝑣).
∆𝑡
Figurev1:Displacement-
TimevGraphvforvKeyvDifference:
Distancevisvalwaysvpositivevorvzero,vwhilevdisplacementvcanvbevpositive,vnegative,vorvzero.vAvroundvtripvyields
vzero vdisplacement vbut va vpositive vdistance.
Uniformv Acceleration:
Whenvaccelerationvisvconstant,vthevfollowingvequationsvofvmotionvapply:
1. 𝑣v =v𝑢v+v𝑎𝑡
2. 𝑠v=v𝑢𝑡v+v1v𝑎𝑡2
2
3. 𝑣2v =v 𝑢2v +v2𝑎𝑠
(𝑢+𝑣)v
4. 𝑠v=v 𝑡
2
Where:
𝑢v=vinitialvvelocity
𝑣v=vfinalvvelocity
𝑎v=vacceleration
𝑠v =vdisplacement
𝑡v =vtime
Figurev 2:Velocity-Timev Graphv forv Uniformv Acceleration
Examplev1:vAvcarvstartsvfromvrestvandvtravelsv0.25vkmvinv25vsvatvconstantvacceleration. vFind:
1
, a) Thevacceleration
2
