Zum Inhalt springen
Zuletzt geändert von Holger Engels am 2026/10/17 14:38
Von Version 53.4
bearbeitet von Holger Engels
am 2026/10/09 14:26
Änderungskommentar: Es gibt keinen Kommentar für diese Version
Auf Version 50.1
bearbeitet von Holger Engels
am 2026/10/09 10:40
Änderungskommentar: Neuen Anhang Rakete.ggb hochladen

Zusammenfassung

Details

Seiteneigenschaften
Inhalt
... ... @@ -38,38 +38,17 @@
38 38  {{/aufgabe}}
39 39  
40 40  {{aufgabe id="Test-Rakete" afb="II" kompetenzen="K3,K4" quelle="Holger Engels" zeit="9"}}
41 -Bei der Erprobung eines neuen Raketenantriebs werden verschiedene Flugprofile getestet. Bereits eingezeichnet in das Geschwindigkeits-Zeit-Diagramm ist ein unrealistisches Flugprofil, bei dem die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
41 +Bei der Erprobung eines neuen Raketenantriebs werden verschiedene Flugprofile getestet. Die Telemetrie wertet dabei aus, wie sich die Geschwindigkeit //v// (in m/s) über die Zeit //t// (in s) verändert (Geschwindigkeits-Zeit-Diagramm). Der Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dabei dem zurückgelegten Weg //s// (in m).
42 42  
43 -[[image:Rakete.svg||width=600]]
44 -Der Flächeninhalt unter dem Graphen der Geschwindigkeit entspricht dem zurückgelegten Weg //s// (in m).
43 +Bereits eingezeichnet ist ein unrealistisches Flugprofil, bei dem die Rakete schlagartig auf ihre Reisegeschwindigkeit kommt und diese die gesamten 20 Sekunden konstant hält.
45 45  
46 -Zeichne für die beiden folgenden Flugprofile jeweils einen möglichen Graphen in das Koordinatensystem ein. Allen drei Flugprofilen ist die Dauer von 20 Sekunden und die Strecke von 600 Metern gemein.
45 +[[image:Rakete.svg||width=600]]
46 +Zeichne für die beiden folgenden Flugprofile jeweils einen möglichen Graphen der Geschwindigkeit //v(t)// in das Koordinatensystem ein. Allen drei Flugprofilen ist die Dauer von 20 Sekunden und die Strecke von 600 Metern gemein.
47 47  (%class=abc%)
48 48  1. Konstante Beschleunigung: Die Rakete startet aus dem Stand (//v(0) = 0//) und beschleunigt für 20 Sekunden mit einer konstanten Rate. Die Geschwindigkeit nimmt also linear zu.
49 49  1. Start-, Flug- und Bremsphase: In einem weiteren Testflug startet die Rakete aus dem Stand (//v(0) = 0//), beschleunigt 8 Sekunden lang mit einer konstanten Rate, fliegt dann 4 Sekunden lang mit konstanter Geschwindigkeit und bremst schließlich mit einer konstanten negativen Rate wieder 8 Sekunden lang bis zum Stillstand (//v(20)=0//) ab.
50 50  {{/aufgabe}}
51 51  
52 -{{aufgabe id="Streifenmethode" afb="II" kompetenzen="" quelle="Holger Engels, Stefanie Walz" zeit=""}}
53 -[[Streifenmethode.svg||width=300 class=right]]Das Schaubild zeigt die Normalparabel {{formula}}f(x)= x^2{{/formula}}. Für //n// Streifen und einer Streifenbreite von {{formula}}\frac{b}{n}{{/formula}} ergibt sich die Untersumme folgendermaßen:
54 -
55 -{{formula}}
56 -\begin{align*}
57 -U_n = \Delta x \cdot f(1\cdot\Delta x) + \Delta x \cdot f(2\cdot\Delta x) + ... + \Delta x \cdot f(n\cdot\Delta x)\\
58 -&= \Delta x \cdot (1\cdot \Delta x)^2 + \Delta x \cdot (2\cdot \Delta x)^2 + ... + \Delta x \cdot (n\cdot \Delta x)^2
59 -\end{align*}
60 -{{/formula}}
61 -
62 -... führt schließlich zu ...
63 -
64 -{{formula}}U_n = b^3\frac16(2+\frac3n+\frac1{n^2}){{/formula}}
65 -
66 -Berechne den Wert der Summe für unterschiedliche //n//.
67 -(%class="border slim"%)
68 -|=n| 4 | 8 | 100
69 -|={{formula}}U_n{{/formula}}|||
70 -Betrachte den Term und begründe, gegen welchen Wert er für {{formula}}n\rightarrow\infty{{/formula}} strebt.
71 -{{/aufgabe}}
72 -
73 73  == Bestimmtes Integral ==
74 74  
75 75  == Eigenschaften ==
Streifenmethode.ggb
Author
... ... @@ -1,1 +1,0 @@
1 -xwiki:XWiki.holgerengels
Größe
... ... @@ -1,1 +1,0 @@
1 -49.8 KB
Inhalt
Streifenmethode.svg
Author
... ... @@ -1,1 +1,0 @@
1 -xwiki:XWiki.holgerengels
Größe
... ... @@ -1,1 +1,0 @@
1 -12.6 KB
Inhalt
... ... @@ -1,1 +1,0 @@
1 -<svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" width="336" height="536"><defs><clipPath id="wTVNkAxZSIOO"><path fill="none" stroke="none" d=" M 0 0 L 336 0 L 336 536 L 0 536 L 0 0 Z"/></clipPath></defs><g transform="scale(1,1)" clip-path="url(#wTVNkAxZSIOO)"><g><rect fill="rgb(255,255,255)" stroke="none" x="0" y="0" width="337" height="536" fill-opacity="1"/><path fill="none" stroke="rgb(180,179,186)" paint-order="fill stroke markers" d=" M 108.5 0.5 L 108.5 536.5 M 166.5 0.5 L 166.5 536.5 M 223.5 0.5 L 223.5 536.5 M 281.5 0.5 L 281.5 536.5" stroke-opacity="1" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10"/><path fill="none" stroke="rgb(180,179,186)" paint-order="fill stroke markers" d=" M 4.5 0.5 L 4.5 536.5 M 15.5 0.5 L 15.5 536.5 M 27.5 0.5 L 27.5 536.5 M 38.5 0.5 L 38.5 536.5 M 62.5 0.5 L 62.5 536.5 M 73.5 0.5 L 73.5 536.5 M 85.5 0.5 L 85.5 536.5 M 96.5 0.5 L 96.5 536.5 M 119.5 0.5 L 119.5 536.5 M 131.5 0.5 L 131.5 536.5 M 142.5 0.5 L 142.5 536.5 M 154.5 0.5 L 154.5 536.5 M 177.5 0.5 L 177.5 536.5 M 189.5 0.5 L 189.5 536.5 M 200.5 0.5 L 200.5 536.5 M 212.5 0.5 L 212.5 536.5 M 235.5 0.5 L 235.5 536.5 M 247.5 0.5 L 247.5 536.5 M 258.5 0.5 L 258.5 536.5 M 270.5 0.5 L 270.5 536.5 M 293.5 0.5 L 293.5 536.5 M 304.5 0.5 L 304.5 536.5 M 316.5 0.5 L 316.5 536.5 M 328.5 0.5 L 328.5 536.5" stroke-opacity="0.23529411764705882" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10"/><path fill="none" stroke="rgb(180,179,186)" paint-order="fill stroke markers" d=" M 0.5 30.5 L 336.5 30.5 M 0.5 30.5 L 336.5 30.5 M 0.5 88.5 L 336.5 88.5 M 0.5 146.5 L 336.5 146.5 M 0.5 203.5 L 336.5 203.5 M 0.5 261.5 L 336.5 261.5 M 0.5 319.5 L 336.5 319.5 M 0.5 377.5 L 336.5 377.5 M 0.5 435.5 L 336.5 435.5" stroke-opacity="1" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10"/><path fill="none" stroke="rgb(180,179,186)" paint-order="fill stroke markers" d=" M 0.5 7.5 L 336.5 7.5 M 0.5 7.5 L 336.5 7.5 M 0.5 18.5 L 336.5 18.5 M 0.5 41.5 L 336.5 41.5 M 0.5 53.5 L 336.5 53.5 M 0.5 65.5 L 336.5 65.5 M 0.5 76.5 L 336.5 76.5 M 0.5 99.5 L 336.5 99.5 M 0.5 111.5 L 336.5 111.5 M 0.5 122.5 L 336.5 122.5 M 0.5 134.5 L 336.5 134.5 M 0.5 157.5 L 336.5 157.5 M 0.5 169.5 L 336.5 169.5 M 0.5 180.5 L 336.5 180.5 M 0.5 192.5 L 336.5 192.5 M 0.5 215.5 L 336.5 215.5 M 0.5 227.5 L 336.5 227.5 M 0.5 238.5 L 336.5 238.5 M 0.5 250.5 L 336.5 250.5 M 0.5 273.5 L 336.5 273.5 M 0.5 284.5 L 336.5 284.5 M 0.5 296.5 L 336.5 296.5 M 0.5 308.5 L 336.5 308.5 M 0.5 331.5 L 336.5 331.5 M 0.5 342.5 L 336.5 342.5 M 0.5 354.5 L 336.5 354.5 M 0.5 365.5 L 336.5 365.5 M 0.5 388.5 L 336.5 388.5 M 0.5 400.5 L 336.5 400.5 M 0.5 412.5 L 336.5 412.5 M 0.5 423.5 L 336.5 423.5 M 0.5 446.5 L 336.5 446.5 M 0.5 458.5 L 336.5 458.5 M 0.5 469.5 L 336.5 469.5 M 0.5 481.5 L 336.5 481.5 M 0.5 504.5 L 336.5 504.5 M 0.5 516.5 L 336.5 516.5 M 0.5 527.5 L 336.5 527.5" stroke-opacity="0.23529411764705882" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10"/><path fill="none" stroke="rgb(28,28,31)" paint-order="fill stroke markers" d=" M 50.5 2.5 L 50.5 536.5" stroke-opacity="1" stroke-miterlimit="10"/><path fill="none" stroke="rgb(28,28,31)" paint-order="fill stroke markers" d=" M 50.5 1.5 L 46.5 5.5" stroke-opacity="1" stroke-miterlimit="10"/><path fill="none" stroke="rgb(28,28,31)" paint-order="fill stroke markers" d=" M 50.5 1.5 L 54.5 5.5" stroke-opacity="1" stroke-miterlimit="10"/><path fill="none" stroke="rgb(28,28,31)" paint-order="fill stroke markers" d=" M 0.5 493.5 L 334.5 493.5" stroke-opacity="1" stroke-miterlimit="10"/><path fill="none" stroke="rgb(28,28,31)" paint-order="fill stroke markers" d=" M 335.5 493.5 L 331.5 489.5" stroke-opacity="1" stroke-miterlimit="10"/><path fill="none" stroke="rgb(28,28,31)" paint-order="fill stroke markers" d=" M 335.5 493.5 L 331.5 497.5" stroke-opacity="1" stroke-miterlimit="10"/><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="100" y="509" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">0.5</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="164" y="509" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">1</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="215" y="509" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">1.5</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="279" y="509" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">2</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="26" y="440" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">0.5</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="36" y="382" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">1</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="26" y="324" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">1.5</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="36" y="266" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">2</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="26" y="208" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">2.5</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="36" y="151" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">3</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="26" y="93" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">3.5</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="36" y="35" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">4</text><text fill="rgb(28,28,31)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="12px" font-style="normal" font-weight="normal" text-decoration="normal" x="36" y="509" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">0</text><path fill="rgb(176,0,32)" stroke="none" paint-order="stroke fill markers" fill-rule="evenodd" d=" M 50.452505072665765 493.0703484733877 L 50.46250507266576 493.0703484733877 L 50.46250507266576 493.0703484733877 M 50.452505072665765 493.0703484733877 L 50.46250507266576 493.0703484733877 L 108.28537228408258 493.0703484733877 L 108.29537228408259 493.0703484733877 M 108.28537228408258 493.0703484733877 L 108.28537228408258 464.1539148676792 L 166.1182394954994 464.1539148676792 L 166.1182394954994 493.0703484733877 M 166.1182394954994 493.0703484733877 L 166.1182394954994 377.40461405055373 L 223.95110670691622 377.40461405055373 L 223.95110670691622 493.0703484733877 M 223.95110670691622 493.0703484733877 L 223.95110670691622 232.82244602201132 L 281.78397391833306 232.82244602201132 L 281.78397391833306 493.0703484733877 L 50.452505072665765 493.0703484733877" fill-opacity="0.09803921568627451"/><path fill="none" stroke="rgb(176,0,32)" paint-order="fill stroke markers" d=" M 50.452505072665765 493.0703484733877 L 50.46250507266576 493.0703484733877 L 50.46250507266576 493.0703484733877 M 50.452505072665765 493.0703484733877 L 50.46250507266576 493.0703484733877 L 108.28537228408258 493.0703484733877 L 108.29537228408259 493.0703484733877 M 108.28537228408258 493.0703484733877 L 108.28537228408258 464.1539148676792 L 166.1182394954994 464.1539148676792 L 166.1182394954994 493.0703484733877 M 166.1182394954994 493.0703484733877 L 166.1182394954994 377.40461405055373 L 223.95110670691622 377.40461405055373 L 223.95110670691622 493.0703484733877 M 223.95110670691622 493.0703484733877 L 223.95110670691622 232.82244602201132 L 281.78397391833306 232.82244602201132 L 281.78397391833306 493.0703484733877 L 50.452505072665765 493.0703484733877" stroke-opacity="0.39215686274509803" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10"/><text fill="rgb(176,0,32)" stroke="none" font-family="geogebra-sans-serif, sans-serif" font-size="16px" font-style="normal" font-weight="normal" text-decoration="normal" x="160" y="477" text-anchor="start" dominant-baseline="alphabetic" fill-opacity="1">a = 1.75</text><path fill="none" stroke="rgb(0,103,88)" paint-order="fill stroke markers" d=" M 50.45250391600842 493.0703484733877 L 52.5 493.0341040705109 M 52.5 493.0341040705109 L 55.125 492.8815958669662 L 57.75 492.60994044952304 L 60.375 492.2191378181814 L 63 491.7091879729413 M 63 491.7091879729413 L 65.625 491.0800909138028 L 68.25 490.3318466407658 L 70.875 489.46445515383033 L 73.5 488.4779164529964 L 76.125 487.372230538264 L 78.75 486.14739740963313 L 81.375 484.8034170671038 L 84 483.34028951067603 M 84 483.34028951067603 L 86.625 481.7580147403498 L 89.25 480.0565927561251 L 91.875 478.23602355800193 L 94.5 476.2963071459803 L 97.125 474.2374435200602 L 99.75 472.0594326802417 L 102.375 469.76227462652463 L 104.99999999999999 467.3459693589092 L 107.625 464.81051687739523 L 110.24999999999999 462.15591718198283 L 112.875 459.382170272672 L 115.5 456.4892761494626 L 118.125 453.4772348123549 L 120.75 450.34604626134865 L 123.375 447.0957104964439 L 126 443.7262275176407 L 128.625 440.2375973249391 L 131.25 436.629819918339 L 133.875 432.9028952978405 L 136.5 429.0568234634435 L 139.12499999999997 425.091604415148 L 141.74999999999997 421.0072381529541 L 144.375 416.80372467686163 L 147 412.48106398687077 L 149.625 408.03925608298147 L 152.25 403.4783009651937 L 154.87499999999997 398.7981986335074 L 157.49999999999997 393.9989490879227 L 160.125 389.0805523284395 L 162.75 384.0430083550579 L 165.375 378.8863171677778 L 167.99999999999997 373.6104787665993 M 167.99999999999997 373.6104787665993 L 170.62499999999997 368.21549315152225 L 173.24999999999997 362.70136032254675 L 175.87499999999997 357.0680802796728 L 178.49999999999997 351.3156530229004 L 181.12499999999997 345.4440785522296 L 183.74999999999997 339.4533568676602 L 186.375 333.34348796919244 L 188.99999999999997 327.1144718568261 L 191.62499999999997 320.76630853056145 L 194.24999999999997 314.2989979903982 L 196.87499999999997 307.7125402363366 L 199.49999999999997 301.00693526837654 L 202.12499999999997 294.18218308651797 L 204.74999999999997 287.23828369076097 L 207.375 280.1752370811054 L 210 272.99304325755145 L 212.625 265.69170222009905 L 215.25 258.27121396874827 L 217.875 250.7315785034989 L 220.49999999999997 243.07279582435112 L 223.12499999999997 235.29486593130486 L 225.74999999999997 227.39778882436013 L 228.37499999999997 219.38156450351698 L 230.99999999999997 211.2461929687753 L 233.62499999999997 202.99167422013522 L 236.24999999999997 194.61800825759667 L 238.87499999999997 186.12519508115963 L 241.49999999999997 177.51323469082416 L 244.12499999999997 168.7821270865902 L 246.75 159.93187226845777 L 249.37499999999997 150.96247023642684 L 251.99999999999997 141.87392099049754 M 251.99999999999997 141.87392099049754 L 254.62499999999997 132.66622453066975 L 257.25 123.33938085694348 L 259.875 113.89338996931878 L 262.5 104.32825186779553 L 265.125 94.64396655237391 L 267.75 84.84053402305375 L 270.375 74.91795427983521 L 271.6875 69.91198420301401 L 273 64.8762273227182 M 273 64.8762273227182 L 274.3125 59.81068363894781 L 275.625 54.71535315170274 L 276.9375 49.59023586098306 L 278.25 44.43533176678875 M 278.25 44.43533176678875 L 279.5625 39.25064086911988 L 280.875 34.03616316797638 M 280.875 34.03616316797638 L 281.78397507499034 30.407406155422734" stroke-opacity="0.8" stroke-linecap="round" stroke-linejoin="round" stroke-miterlimit="10" stroke-width="3.5"/></g></g></svg>