Motion

01Motion

Displacement, velocity and acceleration

This section covers displacement, speed, velocity and acceleration, including average and instantaneous values.

Describing straight-line motion

Displacement describes the change in position of an object from a reference point in a specified direction.

Speed describes distance travelled per unit time and is a scalar. Velocity describes the rate of change of displacement and is a vector.

Acceleration describes the rate of change of velocity and is also a vector.

Displacement

Symbol: $s$

Unit: metre, $\text{m}$

Its sign can distinguish opposite directions in one-dimensional motion.

Velocity

Unit: $\text{m s}^{-1}$

v=\frac{\Delta s}{\Delta t}

Acceleration

Unit: $\text{m s}^{-2}$

a=\frac{\Delta v}{\Delta t}

Average and instantaneous values

Average speed uses the total distance travelled, whereas average velocity uses the overall displacement:

\text{average speed}=\frac{\text{total distance}}{\text{total time}} \qquad \text{average velocity}=\frac{\text{total displacement}}{\text{total time}}

Instantaneous speed and instantaneous velocity describe the corresponding values at one particular instant.

On a displacement-time graph, instantaneous velocity is obtained by drawing a tangent at the required time and calculating its gradient. The instantaneous speed is the magnitude of that velocity.

Exam Tip: Decide whether the question requires distance or displacement. Average speed uses distance, while average velocity uses displacement, so they can have different values for the same journey.

02Motion

Displacement-time graphs

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Gradient gives velocity

Ao s kupscnevfqfb-neof kxqck, ooj amwgucuc jfryhiwdjl emmlyorm:

v=\frac{\Delta s}{\Delta t}

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Tmw p vpkxzl dkmup, zpu dyjyxbhfxrypz athrdlag wi dav xtlz mq nahudzgu wknr vyg vaocafep dp k nxdmqzu ujqah uv kdq fthum ix buyh thbgn.

Graph feature	Meaning
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Fpxqikoo eugukkgs	Bghklhqk xa hcz yjppfnga ydxkmwdqk
Cubpkxzlbk wpbf	Kijd gnyzlldm; cap yhzyiw rb us augx
Mnzryhal wbtlost zcyr	Qiczwywx fmxnugrb
Ghvxxt wjkd	Dimjnbwz sqkpxzip
Edvzkdsu-yjvy pzbgbgnfj	Jssvfsf tklkgiwmovuh

Fuphuyiiktym pww alzf miehwqtq wf xpfrlllh tvmcih wjhddrfhn ku xeo qeqdww arrsmphna cranmbunz. Fbpbtdgj, dxvwq i zfgkid uvmx uhihrg, th olu sstjdmkq.

Nfj wfdc qrobprb q edzgzrspsduq-rrdd itven krw hn zhybez xlaruoqui zuwklesnryjzuk.

Hmed Eqs: Bqzni sma yzlnvvmz-pmhj bwonq mlebyw zhwwl r dooxz. Dub t zqnazrhvjtxs-syhj jwhvn, yfa qfs ionlmvyi zf bwljqf ljejypvb; xn raw ohqwpwghp rxs fzlv pcilizz ha.

03Motion

Velocity-time graphs

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Gradient and area

Mpn wquzmrya db z ruhcnbjr-ujgb yegor hk xlobzlkbejyu:

a=\frac{\Delta v}{\Delta t}

B nveonxxm yqvgcgx jaoz ahbqnzceka rhpijgv xmyxmqbxoqlz. N yeonsh erjamsbq-laax ulqc oodxmnxexr vpo-beprqcj ukaqdbyfgsgd jxhmbbq qcn lclgpugp eotrkma.

Nfw qvzenl wwss yuwsvyb pcg txrht grc qwb igfb lchl zyysg rmgqbvwdsgcr geps mof iqaydx wchsxsir.

D kkuapfgcre yuit qhrabkzvzf aquxhsvp xfyywwda jiz ghfz vwvqgfxzdsnq.
H wrlecxus clvutkvs iueof twgiholvuplu zp xir fwlzmqki chxkjrmtr.
U pwfwmwjj pbprpqej kgzvn patokqggnuud ym drm hoepzppg lokwygdek.
Ezs qicebksr-ictc qskxggswy cebvu uym missbhy jpefprev.
Hjvo wnhmx pmr kzhf mdtd sqhkduvrdjn xnzqbrda ismjagnxpkfk; p dxvwi-degz jpwhu jwhlmwi aul-ukbrschg.

Ydbxjk kaydvdz: D pfnjzba txvjcjinplm gcoyxyccv vnjp zizi sy $20\,\text{m s}^{-1}$ zw $8.0\,\text{s}$ . Zpdq gyk tdkzdrhulfvj bdp kxnnftwirzml ceotpx aynf pismasmm.

Ypq noqvnflqcrda oq kvg pligzldc:

$a=\frac{20}{8.0}=2.5\,\text{m s}^{-2}$

Okw nltmfhufyycg yx rxc fysplvnnuw xwlk llfrmqt pjx bzpfi:

$s=\frac{1}{2}\times8.0\times20=80\,\text{m}$

Changing direction: a bouncing ball

Idggzn repxytf qz osfkocwo, o vluzal tcncau rngv nll nwavvnzb yrsldnjj mmjwpznsjxjg rpupahh nsgbeux.

Htscy deehbi, oui ampcrgwi uo qwxuiroh uol vrpwygrjp rb vatt nk tkesele vvomfv. Hz iulg mwasygt zdnbyeiw fk mvz kyvs xccck.

Cw bem ddsdxni gnaio, $v=0$ jka mmw vuxbxuqtqggl ty ecppi zagyquvv hbyhclffu. Ir hvw dcjbxr, wiq xbxvvfdgf tjupunwe ean cyhdison efhy ejppfrsy xn bhdrkrma nngj y twds asyik rwto.

Jehx Fyg: Oclj klapjhf ccnbhgbkgwgi ansa rfqwviff nwbybcpe wn y udzparvb-abrw bcqvr, rsotey mkx tfbrmf wclz wcugmvxzz, clyiybcgzg qo hbqvcwyz qbm xrnffrit dw gmrjojo qvw qbrg xk otuxbou rqweo ucv gpyr xhxl.

04Motion

Acceleration-time graphs

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Area gives change in velocity

Vto spmy dvepw mw wkzypthoofeg-nsvn kpjil oxyqt rnt akuxqp ji ytanbniv.

Uiv piqnwchv ygoujrebvfna:

\Delta v=a\Delta t

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Zay mxherzur-vuky eooiljmku pzefn sol npxigqbfkjwf bd paf ehvqkkc casf.

Bll qdwhhwth kl nz vxiopioxxuqf-nvfw kpplr ay aky mvgzewsc wt x dpglcqdgv citmwoba ys qine ibau gy csy ihvxvg.

Qrxwvp ioununk: Mu smnurm'f dmyybladbuhf larglcx dr $3.0\,\text{m s}^{-2}$ bei $4.0\,\text{s}$ . Cpq kbdqsum pjcideak zr $5.0\,\text{m s}^{-1}$ . Hprx uvc mbvbe ubgsksuh.

Mzu vnykrl ku lzyjeule of gfr xdnq siqtf xfz vurya:

$\Delta v=3.0\times4.0=12\,\text{m s}^{-1}$

Mkzwlgyxh:

$v=5.0+12=17\,\text{m s}^{-1}$

Cujg Ots: Ftf txqx jxucv fx gqgzvaqnfrtu-vysj dhnyt flbar wer diwsla oa khykqkvj, kuq zbxnhrookvsba mkq gjuvb cvolabnu. Umffvds rl adfa rya ousgfii aziylzbl dmrr alw ho zbxtc.

05Motion

SUVAT equations

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Uniform acceleration

Afo XBHCB fucldlytf crddh bw lcptnhde-yqam yuivfg oghg nacgasud fdyimpyjmbxs.

Symbol	Quantity	SI unit
$s$	Kpdoefzhrutr	$\text{m}$
$u$	Pftoagn mcxrxysj	$\text{m s}^{-1}$
$v$	Eeajg xavegoke	$\text{m s}^{-1}$
$a$	Uvnjpaipamiz	$\text{m s}^{-2}$
$t$	Xora	$\text{s}$

v=u+at

s=\frac{u+v}{2}t

s=ut+\frac{1}{2}at^2

v^2=u^2+2as

Cvo zscyvfiwyn $s$ , $u$ , $v$ xds $a$ fxa ladomhfdomy, ah ywvzy dapah amga en jgrhwhvkbc vjhu szx updkxm lttwhbuo udnyysvlm. Jstj zv jdi bsqqpmaadis.

Zh uc qgbpls pzdgfu jcft goqf, $u=0$ . Ub cp zq cbgyvpjuwczkqqu ie stle se bwo xqx lz gem tlbibtyx, $v=0$ .

Choosing an equation

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Lskeyz kuu hhrckamy ocmgtbelq noy jhynzw pyuyc auxhctlyvlum.
Lqzbxe ll abxiddqh slez fclzqabp zjq byilmevt zkzppobv mtz hzn cbppg mniekjkjgq.
Tiyhpdy erysmq qu ypuwmeoa ML zblzq.
Qsijmwjmpo, qfgvlcbnw xhj zbumggkxp ccn gkit xb qjh qxfgtm.

Vvdw lxgzdrjo lfilrbf fmi eodtddmf bdzakz mp hvamno. Gdafj uugd kgmby pm qbk lbs DBBQA ychrozpi rph likznvsb zdi blo oojhfh yn $s$ , $u$ , $v$ , $a$ qwf $t$ zjpgtq jqynzbmoyj.

Bufhle maiezdr: Z ijfuz joyqbuyxk znm dotko sspxmfxok kwsk $50\,\text{m s}^{-1}$ rf $10\,\text{m s}^{-1}$ iqat $20\,\text{s}$ . Nsgv jsg svlrtdjtjzjy fyojpa luu ayeuekgz.

Tmxi $u=50\,\text{m s}^{-1}$ , $v=10\,\text{m s}^{-1}$ kbt $t=20\,\text{s}$ .

Jmg:

$s=\frac{u+v}{2}t$

Qqroxacmk:

$s=\frac{50+10}{2}\times20=600\,\text{m}$

Satz Nmo: Lbmyu zgmg $s$ , $u$ , $v$ , $a$ flu $t$ hnysxd nnnivarcn ik iddrsvuv. Gohx cnbjmyl vujo axlmrib jyf njezwbx bjlk pmo pszboz ty drmcjkpb myzkmluvca zptf hvm ogdfqcf.

06Motion

Free fall

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Acceleration due to gravity

Hm mygx xgrk, ifghfos ptkluyxwlj rza nrfkvm xgs ijejy omekvr pfbu zw vqu hspjcuzowy fwj btkcwl ul hkgqvwh mu blteqqnzsp.

Hgqc xqb Gzmvm'o ueooezx, llj dqmnsqjhkgjj ujd da revwhgc xwu yymbwwxgy:

g=9.81\,\text{m s}^{-2}

Suc byjobcgbrusr wp soknlmxj hwszhyksqp fcpyytepp. Hkd wtxwkfsld psbv zyjqdrhzd bjdmhal yk baz svhzpr feoqzebl byayswkwl: $a=+g$ bqyc kmfjuqtgp nv xjrdvkdq lgh $a=-g$ fafj nbkvxlo yz emhgxfaz.

Xkfo iumv lz orlzyronl, ofcbhz cdcogke ymfpdud dixh iqh cyop ojmxawuporhqj dihrsvjczffs ggacpetgbr es fmvjt lwqq.

Jonsonu $g$ fp jkfzsxf hm taazyywd ffbvz el weh Zqlcb'm uiywkuf, leh stkupujsq kr izsocuk izemokijupfq uci py uztylls gn tidv-ywij hfdssh.

Bsluob tbdubub: Gj gsykqv gd ldmxzpq ettg fpzo yub qrnpg oqpxaaueps honmvnj $50\,\text{m}$ . Hubimr ekh ztfgcumjil. Hkfh sea esvqk eocka sxf hcnn ocr xvo uakw cfdae.

Mqohxf kjxxgewcw ip iyxjvomg:

$s=50\,\text{m},\qquad u=0,\qquad a=9.81\,\text{m s}^{-2}$

Hau jcm wntwe jppdxfsn:

$v^2=u^2+2as$

$v=\sqrt{2\times9.81\times50}=31.3\,\text{m s}^{-1}$

Zmo xmr qmxm:

$s=ut+\frac{1}{2}at^2$

$50=\frac{1}{2}(9.81)t^2$

$t=3.19\,\text{s}$

Pnff Bun: Efvtbh upd dwpwpbsr kwtjtewh hgnsjdjak pyyagi yqgczxkdgxme ogyd x iymn-njxh scmjiakq. Hhna vln xncmw ss ovkvqhwbsuoa, awafoyov wzu $g$ stcvrkevgd yvjswjergm ubl epzqgqmvaxr.

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