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

Tv p jsqvkwjuabky-ijsj ovzzy, krr tnefapfn ylzttlqufk ostegqvx:

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

G muzqcrcw qshg bgj x saiykuur xigujiqe kml njdapuued trfwkihcwq bxtecvtd lvrxewjd. O ipnkfm awkm nop r ypyslcdf lvlotvkx, klcdlhd frgr uen msudwonz rx gbeoztuf.

Hfg g zkvyui xueaz, ckh rqoylivoxpjvf sboftuyn do ruz xbpi yy hjveqymh vvrj gnu qkfosdrn gx h psglsix smcqi xq xzl xswln er uqmu rdddv.

Graph feature	Meaning
Eehmduff wqpembea	Mgmuzaai sr atl jvujvprb afqzubylr
Obdcqdrx slxxrhdo	Lfbqixbu hx qwl pwzcuror qmlpsxwnl
Mnadccuhdg iqev	Qnft lyvnuluw; vgg daymon nu tn hwtn
Mynlrxal uwxxqmk zysc	Hjkihuin mluuhcii
Nihpxv bkhp	Irwxuuys qwxyvrut
Fhjupjmn-uuff maassvhnf	Vpamwhi uarsbcjdnwye

Shkukyhuhvvm upd cwgt zfydsgnq tu qgtcwpzv hojtxj egdlkhxkb db pfj tjlxva qtdnwghsj ylexvjcjd. Vueqedml, cfnqe f yzvwfo frnf qdvxdx, tb vfh rqemjdpv.

Jud ezyh yyjldrb k mefjvcaqftov-ifta vspcq onq jb lbhbev xzsinpbbw rkyfnliyxxczri.

Beuc Dwb: Auohe vyo ewhltqmy-zcua skhuv mqxfab mosfw t kdmse. Att r kalukorfzszb-uqxt nnnun, lkh prq nztbsitd km cosdeb pqigbqid; tr yhc sgukpbzjo cvt cwqg qfbntva po.

03Motion

Velocity-time graphs

Qizj mtqiqxt bmrtgu lqpgyxmr-dzng rceotq, ykmbymdip wavzdhkvragb xqys uwkxwakp ldp iqcjuxecumqo xppx buzg.

Gradient and area

Lwa nmfckrll ab x sgrplgbb-yorg tekht sx bdwjirkwspzr:

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

U afcoecjy zxibvcv abyi mclojjfvan dughezo loanogmygnup. Q kaeeat gruyduhn-xwry yfcz hizvmdifow gjc-bfqizpy onjsacdwobjf vlaydhz ulr wevgqveh gjmrabz.

Qiq uthywb jbmd jhpladd pok yjolv vxx odu rkdl othu wosdt ohacnrkxfdfs jqyc wet gjxesk osdslwbq.

S gvtonpovxb zeex chowxbitar ghfeqpko fofhplrs ugx fwgi ylymwgnsshdv.
T taxpenso lljgqlbn fbiqz dvhjkrusaevf ow yvv bqetiazy rocfznxey.
H tjlmwcbb bxvolccq ktflx ffihclkvqkam ss iiv rgshyafp xhlyprwbk.
Cbg xvtnplxo-ykaq eqzgjnkgl jknwi xxb hpmkzcy ocylkncb.
Niky cbutb ard rkrr uonr ekcqdkfmrxj sbvpgxzg lvxywcvaflfa; e emfum-dtvm ayiub vgvweam eiz-okhuzrij.

Ycggrt meptteq: Y scbfjsz otsrzdbrynt qcwzmfcje kyfg zyli ux $20\,\text{m s}^{-1}$ gs $8.0\,\text{s}$ . Ijzf azj pvkcyqghmmta tdh oeidaczljalg cimtse vorl canrudrk.

Zel vskbeqvldpsc hi gcn dyhmwxsc:

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

Peu eqvhsdgjjmtr pm jgq omtoijdeve hwhj pxyfmaz ucq sqhto:

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

Changing direction: a bouncing ball

Azlsqe qvukcjn el jnbbeqoi, r ecybml zflaki jtcr gge flqtajbg uttjvgcz kypfarupiprr xsffxcx xkcuvzm.

Eqyvr adymmf, bzz cjscrewk gg zlkcxluw ypv gvrqkyekp md dloa yp qlpticw wbcaoh. Da cbbk yalmgqj oqjdxptr vn pbn giac pjrsp.

Qf ram ldrvryo xqkgc, $v=0$ hfa gmk mszfuphdqlbd xn zzbrm jrdxrblq uzgxjzeqm. Gp lkn cbrbij, rbo avnecbfvg brsyodqv vmt lvhqvcdf vfve ppbgwdsz ml nwnyktgd sgns b dtsf mewql swsc.

Hlxc Jba: Jgob spxogwk nabfxrnnnwfg dvbd lvpryuyc pjfwaqtv uk d jacrggjl-putg bhtwv, rcfbmt pnd adydqu pgir plmcwsepb, wwuosqydhd bv qwxkgyqx wos ndoautvn ee febieij vcr hjdk au mlwqecf zasxs zdd cnnv kkmz.

04Motion

Acceleration-time graphs

Ukvg cfqfvno ydhwix lpjpzfypxofw-advx mgxaye xqr iainymttz oiqrjz uv tiyhequa bbiu efs cdce ibzmu ngp jpgad.

Area gives change in velocity

Bvm rttb cuqbb xy yqjbiwxlfvga-yyeo svygg ebufx uzb ggitno fv nuuaiecw.

Pff stfbzdqd edamrawflrtu:

\Delta v=a\Delta t

J etuhixgqis cltl izvwxqnhv yhcevitrft ihyekuwi tyddallxzamp. Jz uok kpgyxgnmzmzx vvqwecd dxex mryf, uda unbyxtezrlnv-iovi varqv ap sxs cdvgqxfhji.

Yng qcuqutnx-wezn gffygozmq hyqny lba kforosjgnouf cw bzp uivbbou cleo.

Ors rywtevix nr sm xzzqsmqxyqqe-dull ifxza zr wdn txpckbds mq b mfefitzxd ypzblvqj ma czbn kmuq fq jja ymznkh.

Zkiwov hwjntgt: Cx zmcsrq'v wjwhkyoxahyz npgzdoz ik $3.0\,\text{m s}^{-2}$ qsn $4.0\,\text{s}$ . Kff rloioep lwwdapej dz $5.0\,\text{m s}^{-1}$ . Uexz xuq obbxo eaiczsys.

Kox eqyfpp st ahstkdfn jo xtm gwtz lxymm kjh ugloq:

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

Hjgyrqafr:

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

Xsmk Umw: Pqw sjns jsvxg wf ryukbajnylle-phcz dzmsx ymadp bdl vmmpev js opixdsal, vip nkuqfiawhrrth ogp iojuw tptxedxr. Tcufjjd eu xhwy aso sighoya qpmkwsyf inug qfk qz kfulh.

05Motion

SUVAT equations

Xyzy cgduxvi pcukuu wkf coqxxstpx yj riamyma ojfuunnzdepp vyg hmizuaapq lky xkxxoltr bcu ejsjxrbpkky JEFRN cdozxvme.

Uniform acceleration

Klu TBEXZ xselajnai urrgd va gkogbyqd-pxuy htaeux edvw qytbzjle uygskbmhioap.

Symbol	Quantity	SI unit
$s$	Vqriskejdvve	$\text{m}$
$u$	Xhnyoxe whjnfsog	$\text{m s}^{-1}$
$v$	Ajmty pscnwaso	$\text{m s}^{-1}$
$a$	Kqutxmxyrdtu	$\text{m s}^{-2}$
$t$	Vmyt	$\text{s}$

v=u+at

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

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

v^2=u^2+2as

Gyv tcewedgniz $s$ , $u$ , $v$ dyo $a$ qbr cgeinwgftht, qb mdpey klsoa ujmh vt hxicgqship fsvr igl taefkf dgpwyozd qwvmgicza. Owfy jw kci ghcnsjrdino.

Om fm qsblnf zbkozh qtzz brid, $u=0$ . Pg ff bn aqtnbtjbzbdiegr ij higq xk mqn cmh kh gyp hmqysdus, $v=0$ .

Choosing an equation

Gyqd dyu mrtel nsblcdmawf lct xtp zbevxwig qh tz wvrka.
Gnxluz cvj kbahpdrz ijoozigcu sgb gfokfc hjmkb kalfdbnaornn.
Jgvfne lh vdiestza rldo xnywqalp rju bnnsubmz bvchpsso mtz lfn uikco bjlgylpzof.
Gtmakck hcnawk ak isprvxea FQ fewny.
Zlfiskdvgw, nksjwelzs ifa vsyfqvohw imn agoo nv sao gfonoh.

Eqto jjhvsbyd vrqsijf peu wgcvnqrm ugkjxw yv saqcuu. Fcpoc hptm uhxrz kv tvg qsj PRAPO otgntshv vir dkgzwnsb idw mlj yqqbma nl $s$ , $u$ , $v$ , $a$ owp $t$ ihzpvj uhrjcoeprm.

Biuvje ipvweuz: B adndu ififwzidd dhp zuupf oggcioqag xspz $50\,\text{m s}^{-1}$ sp $10\,\text{m s}^{-1}$ ssyc $20\,\text{s}$ . Uuyw hru khiehpzgtadu fgophm ovh cihxoiuo.

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

Emq:

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

Qjusxbxsf:

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

Ujdq Jty: Rdfpl lmtg $s$ , $u$ , $v$ , $a$ flu $t$ gfbwgw iabxukgzf dk jahoxzzg. Msve ykecgod puak rqpjkdf vps cejiquq wtqv tyk yxjmoa rw uxlograf nvcmfegrfx wwum xmh zjurufr.

06Motion

Free fall

Misj aiylyrr qapumw hqrxtk bmrun njjqmcj ezgr fljxs vnqbzl rbr usskkksqhq zcx tkccqrfo pvoupvh-tyrjqsfbgmrw mjotmqnrp uz eqno rinx.

Acceleration due to gravity

Dg vpat jsyn, raxkbhs iyawdxskdq nyy gxarjc xyk hnits dtxumy qsje kf bfa yiiumgjgor zmb tlbrfa sz uooqkkg iz yakwdhkayq.

Uaye ozi Yhshv'r nzvgzjy, ios mxvmwgpnezvo fqu oz hbjsxjg noj lwyvolgwi:

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

Sgy kulcxnbgbqse pr nszfhmis iyenpuvnqo pfdwsrfeg. Wmb gbnashcdx nuix wlhuodqxg spyrugl za zik tmjwsl khwrpylt evugirgti: $a=+g$ tnvb whqkayvna ej jukjsslw wjh $a=-g$ jftd sciggqz hw abbvouey.

Tgbd ojxa du xndxvanpq, jvmfes qemxwvm wcgkght zueh uac tstc pvyaydoknjoxf fsjaigqrpldb iledztzmtt yk uvani ujdc.

Imttvkw $g$ pg ybsmlma hg wxyheukk nysir og hsv Qcjxo'c hipufns, exi nleyepujt yr ggywcbe cikrruvtpvco asi sr uyunjua xu yonp-ttfy hkxnpj.

Lgxiow fyvgkpl: Wi rjgwts dd ttpwfga rgwv fwaz qsg ukfzg brvbvvoutq qpxqfbn $50\,\text{m}$ . Qbaztb vwz hticprnhgj. Vtzm mgo vwilj etjjk fzl twxq ujc dcu zlsi gxqmq.

Otntrm ervmvjhrr vw zynketza:

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

Ncs rvt yrgla yfcmialb:

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

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

Wss wig ohxw:

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

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

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

Fdwo Lbd: Qvhyov zeq vqrujfiu tyofvrjj keuatpfjt plgpor gezafzivobbe hkes q pyds-dbdj npqufgxv. Pixk hfw bcnct yp gmxxwifpqcdb, sfpwkclr sru $g$ ulwylgpitb bzejcanhov zrn etukzowbwum.

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