From 990a3f22a94c5170b86016dcb4d3b592a1bca58c Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sat, 27 Jul 2019 17:44:41 +1000 Subject: [PATCH] [spec] re-render graphs \& add kinematics formulas --- spec/spec-collated.pdf | Bin 376948 -> 377933 bytes spec/spec-collated.tex | 23 ++++++++++++++++++++--- 2 files changed, 20 insertions(+), 3 deletions(-) diff --git a/spec/spec-collated.pdf b/spec/spec-collated.pdf index 15ea285a1651a87c98613586dc77ed5aff9b47a1..2af5693473797d195b23dfaba30384137b9a4b4f 100644 GIT binary patch delta 22239 zcmV(%K;pmjfEUe(7l5<@P;CS=G&nVrF(xQ~&0AS>+eQ+8*RSBd?2?sWGWU_Heel}O zu9G->Se2^H@c@&ML@!0kBqdAv_1irIfEZu^k`g2536KEx%zV?w*WCj;x*XBb-_QPb ze)iQjcI1fDT#wE#MoJUy#K(E6#SY}I{xm>53k?8 zz~Ro{ynGjg{5f3SzJB?S7eC;CLE72t^Rtg<41P{Wte!Fj`BaXk*JnTfN=Gwz_U(ug z;q2&>fAe}IEhE;!?PB!q?B8d!2?95NT@;K_gbNua`OU=X@$GWDov)TKCo9LR3%rW) z_NoXsvjRX7d|WJN#c~2S+wlLp2_TcN=lT0Z5zuHtPDdG+L}(Mnd2@%~>V@6^2m2 zL%_sDb9phKOj@FhI0d38N?X%{G_epSVNNlkun;$9O&YpuqampMrd%#yKIWYZNR&)B7-HA!Ugh^;^q-LPl zqNTO54DOnm&$=LiYoNA?5t(Gxxbd6i1pM!|xLoI3?^Q+_v&xT`F+*H6^Y`9QueMj< zi?khY=Cfjhw{E;$h3jUuxIG1ag?Ite=Zjf;Xyu|rHmVYoTQPeTEw)w z0+@`gpL^#C!dA;-Hz88ZUl3DGzufHC_c!Zevq^abFkxGRTO`njofZzD(@YIaPJjhv zjhFIG+J$jZ`2SO)f=;q5r+;A|iZ6XH%)9$$b?sV93SO=qj<|Np0bmIU<5<3$| zKUBMn66wmht!`5oGw@OZBdV}%h+{iu7DU2*CKbS4vbJ}{O}@@6ZYWA_D3SWfiu!FH z$^F1zfNR0{EeE)tAy&bKg0ldJ_7v_{Nuq=+F%Z2u(L6*ZJ;42f0hW&OB1PPsI)?k* zr$lHcyHT_wLk&}ZQ0jgM4vNVOa?r24DwhF!V;nSqgWZ%6{*;1{0niv01yTBQ2y9+6ozG3N2Tpthd1?r+9t4DAs@31UGbDY_C>-J|~?8e{`{0hbKt-YVPss zbq~Sf&UiYex%oX65BT1$8t_FzKnPXk3sS+qw2J{J2HwdU zFP#j`P)G`YS`MCfbDb|13Bd{sS*Zbq%9w`tYsl|zL=Z%Ia@7p>N7X6zRZ3$qzPit+vO$K=QG|pY3LFzNG;r(^uWnZ+ z1&bbkH(wUlz%{4nKXB%7P`Y ziv`T3n6HZ**TKsIRBs;bBC$=r))DWrTnp1v1AH*$x3n=VUry<cUwgAKdS^(MrAU-Jp#M?B05Kb+j+{E}Ka0GA4YGPdB=TjDk z*|&6%>XZ+gP6qVeCeua@QNn^$Q}N$!W)C6U>SlE>RhfXSS;ytQWSJj~7Yu<}ivtdS zr4hiRq#I&f;INewBgnyttcFb0C14~hH+TZqoPDe`(igtfP=3vbVZ4ddU1EU-W~J>R z3xq#?9kX|V2C`MG3Z)j<$^%uD!t_=LYL)dfN0K4bo&|9V$bm*3v>plp_AK5GBXp=CvUV}grfYeozu%&OGl9YHUjc*| z+seEnpH&HJ&Yvgi4$PP)_;-CCp~2C3qL>bW6#v}iIibo|`|ZAKw0>d0*IG3);Fp~n z>^C{=LhybBBp@I;Hk&!P?-GIKs*8C&CWKV2^|52NQhP@@STXj?9E%ZcB^644wre^4 zZ2vYXVQ1ibQ~Mm5D6=kjY4F<`v^x$yJ^OBaB+$LS)DSLA2n71b<-|i7D)tNo+XJq> zIn);+KjCSAfgFt1y(i%QPKFQ*#Fdgd_?C6BO|thSY-^WHoZuK36kDkM#Wc_#us}94 z;3AY5tRRCR)3WL2aJ3|?V6;qsu~kgdrddBeB&A9!sif@S%=b2#`O-t`$8tv`VV5vr zDt8H1B3P`Z2X51l_N9U<#BnBDYi(^cQ*KYUTrvkyz(LN!iKgZWTU|-g4wb&&d*fE? z=2quPN@TU%N9Z{BA)v|V0&U0);OkDl3wPM1t{88eJyGxaI~B#wcGv@dwgX9*hG3$H zqv-+L;UyUi%X{|2G!GT@?MKSj_3npmSXerCX#;vPd`N45?6RP>pln)dUm)#^3P9J8 z$Ut;;Wq`}E)^T^4Jcxo#v&O4lF}ma$cA15PZ;k3jdp9AQwm3aUm>i@Al7~`QA2R@M z=de*5T{I$*p9Oh@dQ&STK| zE!hFnuxAc61%kM9?fQy8p1BTokM%KMUkU$yAC7AJve_s9tUf3h=S_Ib#0{7hP*klB z4=8!UAiy?74cVX4$B&FS-5lO0pp?Kg>leOGc0{Cab)XAzRT24^{%*9vK})iRF1-(7 z91=7ve1qjPBV)*a;z6hzaR5%sMyPL7S`PLtM@m9C6oaM@aS2IpZ2L`@Rf~c^ckEf1 zM9E1~>N$ZD985|)sRjV-7hvTh#qpG|9XLnC2<7^Sb3}ly+Lz{(2WKL8(D4S3J42K> z<|3sc3MlJgxJhn?aDym=&6WMQQSa^#Kf2bzt=&blG6t@HC#qM)lW1lw2MEkcQFMSn z{i>Ms?rOE(7Mt?;`K;j%<$&LlF7kVd?6RtLDrr*osGenlVWRIg5k;e^1F+IGso5?; zi;yt>$bRKyt7>+%CfH}j?4C90V|tT4YqHUrl81V+EWLkG5r9dj%#r(`&RN1_f0sJ; z{~uD|0OK-$e{gI%jd#~gzoJD&)acn3#UH<@Xs{2dE^fvYB?et3SZICp@yCgi<6^q? z2Oo?YUzA^}Y!aXt=|c~;ubgHl$0q9Od2HfFeNV>+xHa2#Q127-j#}sQR6uI9zxFcI zhvW#2qgfwP^jC3t@Cd&pN1&$ms{b2oGBQm2P_w{)w83#7+-9uDuTGQ3ejSu&$Mc?T z+q;@>;_3-!$!eFi^*C|fok3nY{xrwW8t@L=-e1D4P~nT8@K5|O3vbFkC*H0uN}4Gk z=C}I3#d1~-QnunZpG*Ga+%+ z`iKKQC$pb^N1+Tj3iJ3K`cHSw@diQk;riKC{S+H?!1>!#+GQ(JYdp3y9H^)HVtTu{ z-6Y6g_&7jSjNw(Wo`=r~@JtaB_rC85lq<7;kME0ZDg@w{h@W9QCF_a&uGs>6f=NAM z$0A@N8A*a+sTd0X$g=t8QN8}Op=TUmq-v-Ji@gZZ9EVIY?KH9q*h$m)t}r1R>n-Zp zour>v`XCcXWc;r031{{df8 z`6h>-cmao>cmju?cmubecm(a65HvChFHB`_XLM*XATcmDG?(E%1Qi7_F*h(Xm+^Q3 zD1WyFP+RR5E{qf@P~4>icXxMpmtp||q(DMRa4il+ij~shQmnYUTk%5iBE_M&`{k2! z&UeoL&)hqi+1bzXwccmFGn;`%TaQD=7HSPrghJpPJe=I3067hP6&`K?H#Z+AH#aW^ z1A{&o?gaXm9D~6SyI!`V^)CJ%0Zo-K@d0$PU46(KQ zqYYa(7cK(`_>CJ#MgDJ_ClSVfWcDC9K!BT@TbNfE0D1!edD=K|{gFW5%YOy*r;_In z@ly|eJ}yuffZbCYAV07j=;;r`2L^Np0pPA~AU~gfD*hW`@bCa^!8UM!HOL+e!T3+~ zCo#zGFZ^`=u3%5VEAA)y@c_90`2G9M{E1<

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\textbf{Displacement} - relative to starting point + \end{itemize} + + \subsubsection*{Velocity-time graphs} + + \begin{itemize} + \item Displacement: \textit{signed} area between graph and \(t\) axis + \item Distance travelled: \textit{total} area between graph and \(t\) axis + \end{itemize} + + \[ \text{acceleration} = \frac{d^2x}{dt^2} = \frac{dv}{dt} = v\frac{dv}{dx} = \frac{d}{dx}\left(\frac{1}{2}v^2\right) \] + \begin{center} \renewcommand{\arraystretch}{1} - \begin{tabular}{ l r } % TODO need to fix centering here + \begin{tabular}{ l r } \hline & no \\ \hline \(v=u+at\) & \(x\) \\ \(v^2 = u^2+2as\) & \(t\) \\ @@ -1179,9 +1193,12 @@ &= \sqrt{v_x^2 + v_y^2 + v_z^2} \end{align*} - \textbf{Distance travelled between \(t=a \rightarrow t=b\):} + \noindent \textbf{Distance travelled between \(t=a \rightarrow t=b\):} \[= \int^b_a \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \cdot dt \] - + + \noindent \textbf{Shortest distance between \(\boldsymbol{r}(t_0)\) and \(\boldsymbol{r}(t_1)\):} + \[ = |\boldsymbol{r}(t_1) - \boldsymbol{r}(t_2)| \] + \subsection*{Vector functions} \[ \boldsymbol{r}(t) = x \boldsymbol{i} + y \boldsymbol{j} + z \boldsymbol{k} \] -- 2.47.1