File:Graphs of speed and displacement during braking with constant deceleration.svg

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Graphs of speed and displacement during braking with constant deceleration

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Description	Deutsch: Funktionsgraphen (im kartesischen Koordinatensystem) der zeitlichen Verläufe von Geschwindigkeit $v$ (oberes Diagramm) und dabei zurückgelegter Wegstrecke $s$ (unteres Diagramm) zur Veranschaulichung des Bremswegs für die drei folgenden Fälle: Kurven (I): Bremsen (mit konstanter Bremsverzögerung $a>0$ ) von der Anfangsgeschwindigkeit $v_{0}=v(0)$ (zum Zeitpunkt $t_{0}=0$ ) bis zum Stillstand führt nach verstrichener Bremszeit $t_{\text{B}}$ zum Bremsweg $s_{\text{B}}=s(t_{\text{B}})$ : $v(t)={\begin{cases}v_{0}-a\cdot t,&{\text{wenn }}0\leq t<t_{\text{B}}{\text{,}}\\0,&{\text{wenn }}t\geq t_{\text{B}}{\text{,}}\end{cases}}$ mit $t_{\text{B}}={\frac {v_{0}}{a}}$ $s(t)={\begin{cases}v_{0}\cdot t-{\frac {1}{2}}\cdot a\cdot t^{2},&{\text{wenn }}0\leq t<t_{\text{B}}{\text{,}}\\s_{\text{B}},&{\text{wenn }}t\geq t_{\text{B}}{\text{,}}\end{cases}}$ mit $s_{\text{B}}={\frac {v_{0}^{2}}{2\cdot a}}$ Kurven (II): Bremsen (mit konstanter Bremsverzögerung $a>0$ ) von der Anfangsgeschwindigkeit $v_{0}=v(0)$ (zum Zeitpunkt $t_{0}=0$ ) bis zur Endgeschwindigkeit $v_{1}=v(t_{1})$ (mit $0\leq v_{1}<v_{0}$ ) führt nach verstrichener Bremszeit $t_{1}$ zum Bremsweg $s_{1}=s(t_{1})$ : $v(t)={\begin{cases}v_{0}-a\cdot t,&{\text{wenn }}0\leq t<t_{1}{\text{,}}\\v_{1},&{\text{wenn }}t\geq t_{1}{\text{,}}\end{cases}}$ mit $t_{1}={\frac {v_{0}-v_{1}}{a}}$ $s(t)={\begin{cases}v_{0}\cdot t-{\frac {1}{2}}\cdot a\cdot t^{2},&{\text{wenn }}0\leq t<t_{1}{\text{,}}\\s_{1}+v_{1}\cdot (t-t_{1}),&{\text{wenn }}t\geq t_{1}{\text{,}}\end{cases}}$ mit $s_{1}=v_{0}\cdot t_{1}-{\frac {1}{2}}\cdot a\cdot t_{1}^{2}={\frac {v_{0}^{2}-v_{1}^{2}}{2\cdot a}}$ Kurven (III): Zum Vergleich: Fahren mit konstanter Geschwindigkeit $v_{0}$ ohne jegliche Verzögerung oder Beschleunigung. Hier kann kein Bremsweg angegeben werden. $v(t)=v_{0}$ für $t\geq 0$ $s(t)=v_{0}\cdot t$ für $t\geq 0$ English: Graphs (in Cartesian coordinate system) of speed $v$ (upper diagram) and the distance $s$ travelled (lower diagram) as functions of time to illustrate the braking distance for the three following cases: Plots (I): Braking (with constant braking deceleration $a>0$ ) from initial speed $v_{0}=v(0)$ (at time $t_{0}=0$ ) to stop leads to the braking distance $s_{\text{B}}=s(t_{\text{B}})$ after elapsed braking time $t_{\text{B}}$ : $v(t)={\begin{cases}v_{0}-a\cdot t,&{\text{if }}0\leq t<t_{\text{B}}{\text{,}}\\0,&{\text{if }}t\geq t_{\text{B}}{\text{,}}\end{cases}}$ with $t_{\text{B}}={\frac {v_{0}}{a}}$ $s(t)={\begin{cases}v_{0}\cdot t-{\frac {1}{2}}\cdot a\cdot t^{2},&{\text{if }}0\leq t<t_{\text{B}}{\text{,}}\\s_{\text{B}},&{\text{if }}t\geq t_{\text{B}}{\text{,}}\end{cases}}$ with $s_{\text{B}}={\frac {v_{0}^{2}}{2\cdot a}}$ Plots (II): Braking (with constant braking deceleration $a>0$ ) from initial speed $v_{0}=v(0)$ (at time $t_{0}=0$ ) to final speed $v_{1}=v(t_{1})$ (with $0\leq v_{1}<v_{0}$ ) leads to the braking distance $s_{1}=s(t_{1})$ after elapsed braking time $t_{1}$ : $v(t)={\begin{cases}v_{0}-a\cdot t,&{\text{if }}0\leq t<t_{1}{\text{,}}\\v_{1},&{\text{if }}t\geq t_{1}{\text{,}}\end{cases}}$ with $t_{1}={\frac {v_{0}-v_{1}}{a}}$ $s(t)={\begin{cases}v_{0}\cdot t-{\frac {1}{2}}\cdot a\cdot t^{2},&{\text{if }}0\leq t<t_{1}{\text{,}}\\s_{1}+v_{1}\cdot (t-t_{1}),&{\text{if }}t\geq t_{1}{\text{,}}\end{cases}}$ with $s_{1}=v_{0}\cdot t_{1}-{\frac {1}{2}}\cdot a\cdot t_{1}^{2}={\frac {v_{0}^{2}-v_{1}^{2}}{2\cdot a}}$ Plots (III): For comparison: Driving at constant speed $v_{0}$ without any deceleration or acceleration. No braking distance can be specified here. $v(t)=v_{0}$ for $t\geq 0$ $s(t)=v_{0}\cdot t$ for $t\geq 0$
Date	25 October 2020
Source	Own work
Author	SweetWood
SVG development InfoField	The SVG code is valid. This vector image was created with Asymptote (dvisvgm used to convert PDF to SVG)
Source code InfoField	Asymptote code // created with Asymptote 2.67: The Vector Graphics Language // using online-editor: http://asymptote.ualberta.ca/ import graph; /* given / real v0 = 10; // [m/s]; initial velocity real v1 = 4; // [m/s]; velocity after deceleration real a = 2; // [m/s^2]; deceleration real t0 = 0; // [s]; time when deceleration starts real xext = 0.1; // factor for extension of x-axis real yext = 0.15; // factor for extension of y-axis / calculation / real tB = t0 + v0 / a; // 5 s; time when deceleration to 0 ends real t1 = t0 + (v0 - v1) / a; // 3 s; time when deceleration to v1 ends real sB = v0^2 / (2 a); // 25 m; braking distance for deceleration to 0 real s1 = v0 * (t1 - t0) - a / 2 * (t1 - t0)^2; // 21 m; braking distance for deceleration to v1 /* Picture 1 / picture pic1; unitsize(pic1, 30, 10); // velocity at time t of constant deceleration with a from v0 to v1 starting at t0 real v(real t, real t0=0, real v0, real v1=0, real a) { if (t < t0) { return v0; // before deceleration } else { real t1 = t0 + (v0 - v1) / a; // time of end of deceleration if (t <= t1) { return v0 - a (t - t0); // during deceleration } else { return v1; // after deceleration } } } real v_0B(real t) { return v(t, t0, v0, 0, a); } // definition of function (I): velocity for deceleration from v0 to 0 real v_01(real t) { return v(t, t0, v0, v1, a); } // definition of function (II): velocity for deceleration from v0 to v1 real v_00(real t) { return v(t, t0, v0, v0, a); } // definition of function (III): constant speed v0 draw(pic1, graph(v_0B, t0, tB+xext(tB-t0), 11), heavycyan, "(I)"); // plot graph of function (I) draw(pic1, graph(v_01, t0, tB+xext(tB-t0)), blue+longdashed, "(II)"); // plot graph of function (II) draw(pic1, graph(v_00, t0, tB+xext(tB-t0)), deepgreen+dashed, "(III)"); // plot graph of function (III) draw(pic1, (t0,v1)--(t1,v1), red+Dotted); // horizontal guideline at v1 draw(pic1, (t1,0)--(t1,v1), red+Dotted); // vertical guideline at t1 xaxis(pic1, "$t$", xmax=tB+xext(tB-t0), Arrow); // draw x-axis yaxis(pic1, "$v$", ymax=(1+yext)v0, Arrow); // draw y-axis //yaxis(pic1, rotate(0)"$v(t)$", ymax=(1+yext)v0, Arrow); // draw y-axis labelx(pic1, "$0$", t0, SE); // label x-axis: 0 labelx(pic1, "$t_1$", t1, blue); // label x-axis: t1 labelx(pic1, "$t_\mathrm B$", tB, heavycyan); // label x-axis: tB labely(pic1, "$v_0$", v0, W); // label y-axis: v0 labely(pic1, "$v_1$", v1, blue); // label y-axis: v1 labely(pic1, "0", 0, NW); // label y-axis: 0 //add(pic1, scale(0.6)legend(pic1), point(pic1,N), 10SE, UnFill); // add legend label(pic1, "(I)", (tB,v_0B(tB)), 2N, heavycyan); // label function (I) label(pic1, "(II)", (tB,v_01(tB)), N, blue); // label function (II) label(pic1, "(III)", (tB,v_00(tB)), S, deepgreen); // label function (III) /* Picture 2 / picture pic2; unitsize(pic2, 30, 3); // displacement at time t in case of constant deceleration with a from v0 to v1 starting at t0 real s(real t, real t0=0, real v0, real v1=0, real a) { if (t < t0) { return v0 (t - t0); // before deceleration } else { real t1 = t0 + (v0 - v1) / a; // time of end of deceleration if (t <= t1) { return v0 * (t - t0) - a / 2 * (t - t0)^2; // during deceleration } else { real s1 = v0 * (t1 - t0) - a / 2 * (t1 - t0)^2; return s1 + v1 * (t - t1); // after deceleration } } } real s_0B(real t) { return s(t, t0, v0, 0, a); } // definition of function (I): displacement for deceleration from v0 to 0 real s_01(real t) { return s(t, t0, v0, v1, a); } // definition of function (II): displacement for deceleration from v0 to v1 real s_00(real t) { return s(t, t0, v0, v0, a); } // definition of function (III): displacement for constant speed v0 draw(pic2, graph(s_0B, t0, tB+xext(tB-t0)), heavycyan, "(I)"); // plot graph of function (I) draw(pic2, graph(s_01, t0, tB+xext(tB-t0)), blue+longdashed, "(II)"); // plot graph of function (II) draw(pic2, graph(s_00, t0, tB+xext(tB-t0)), deepgreen+dashed, "(III)"); // plot graph of function (III) draw(pic2, (t0,v0(tB-t0))--(tB,v0(tB-t0)), red+Dotted); // horizontal guideline at v0tB draw(pic2, (t0,v0(t1-t0))--(t1,v0(t1-t0)), red+Dotted); // horizontal guideline at v0t1 draw(pic2, (t0,sB)--(tB,sB) ,red+Dotted); // horizontal guideline at sB draw(pic2, (t0,s1)--(t1,s1), red+Dotted); // horizontal guideline at s1 draw(pic2, (tB,0)--(tB,v0(tB-t0)), red+Dotted); // vertical guideline at tB draw(pic2, (t1,0)--(t1,v0(t1-t0)), red+Dotted); // vertical guideline at t1 xaxis(pic2, "$t$", xmax=tB+xext(tB-t0), Arrow); // draw x-axis yaxis(pic2, "$s$", ymax=(1+yext)v0(tB-t0), Arrow); // draw y-axis labelx(pic2, "$0$", t0, SE); // label x-axis: 0 labelx(pic2, "$t_1$", t1, blue); // label x-axis: t1 labelx(pic2, "$t_\mathrm B$", tB, heavycyan); // label x-axis: tB labely(pic2, "$v_0 \cdot t_\mathrm B$", v0(tB-t0)); // label y-axis: v0tB labely(pic2, "$v_0 \cdot t_1$", v0(t1-t0)); // label y-axis: v0t1 labely(pic2, "$s_\mathrm B$", sB, heavycyan); // label y-axis: sB labely(pic2, "$s_1$", s1, blue); // label y-axis: s1 labely(pic2, "0", 0, NW); // label y-axis: 0 label(pic2, "(I)", (tB,s_0B(tB)), SW, heavycyan); // label function (I) label(pic2, "(II)", (tB,s_01(tB)), NW, blue); // label function (II) label(pic2, "(III)", (tB,s_00(tB)), NW, deepgreen); // label function (III) /* arranging pictures */ add(pic1.fit(), (0,4), NW); // fit pic1 to NW add(pic2.fit(), (0,-4), SW); // fit pic2 to SW

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	The person who associated a work with this deed has dedicated the work to the public domain by waiving all of their rights to the work worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law. You can copy, modify, distribute and perform the work, even for commercial purposes, all without asking permission. CC0falsefalse

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	Date/Time	Dimensions	User	Comment
current	09:33, 26 October 2020	251 × 415 (26 KB)	SweetWood (talk \| contribs)	improved coloring
	17:02, 25 October 2020	251 × 415 (26 KB)	SweetWood (talk \| contribs)	slight improvements
	12:05, 25 October 2020	251 × 415 (26 KB)	SweetWood (talk \| contribs)	Uploaded own work with UploadWizard

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File:Graphs of speed and displacement during braking with constant deceleration.svg

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