Sind wir schon da? Sind wir schon daaahaaa? Wie langsam faehrt diese Bahn denn schon wieder... und wie finde ich das als Fahrgast heraus, wenn es nirgendwo angezeigt wird und der Zugbegleiter die Frage nicht nochmal beantworten will?

Geschwindigkeit, Zuege und Zeiten

Geschwindigkeit ist Weg pro Zeit. Also braucht es zum Ermitteln der Reisegeschwindigkeit sowohl eine Ermittlung des zurueckgelegten Weges wie auch der hierfuer zurueckgelegten Zeit. Letzteres sollte heute jedes Handy schaffen: Eine Stoppuhrfunktion haben 'se doch irgendwie alle. Bleibt nur noch die Frage, zwischen was fuer Ereignissen man denn die Zeit erfassen will. Mein Vorschlag: Hektometertafeln http://de.wikipedia.org/wiki/Hektometertafel! Die stehen an jeder Bahnstrecke herum und sind im Abstand von 200 Metern aufgestellt. Als Fahrgast laesst sich also die Zeit erfassen, die fuer das Zuruecklegen von 0.2 km benoetigt werden. Also ergibt sich

v = s/t, v = 0,2/t_s [km/s]
Nun ist Kilometer pro Sekunde zwar eine valide Geschwindigkeitseinheit, aber der Grossteil der Reisenden wird sich wohl doch eher mit Angaben in Kilometern pro Stunde vertraut sein. Also sollte man die Konstante ein wenig frisieren.. Mit der Erkenntnis, dass eine Stunde aus 3600 Sekunden besteht, ergibt sich
v = 720 / t_s
Da das Teilen von 720 nicht notwendigerweise jedem im Blut liegt, kann man sich eine kleine Tabelle dazu vorbereiten. Die findet sich ganz weit unten in diesem Artikel. Vorher muss noch ueber ein Problem geredet werden.

Messungenauigkeit und Reaktionszeiten

Plottet man sich einmal den oben ermittelten Zusammenhang zwischen verstrichener Zeit zwischen zwei Hektometertafeln und Fortbewegungsgeschwindigkeit, sieht das ungefaehr so aus:

Und nun noch einmal in einem Ausschnitt, der etwas ... bahnpraktischer ... ist.

Es gibt da ein Problem. Die Zeitmessung. Wie genau kann ein Zugreisender das Intervall zwischen zwei Hektometertafeln wirklich messen? Laut http://de.wikipedia.org/wiki/Zeitfehler kann ein geuebter Messer beim Sport mit einer Stoppuhr auf Werte um 0,2-0,1s kommen. Setzen wir diesen geuebten Messer ans Zugfenster und lassen ihn mit einem Fehler von 0,2s messen. Zu bedenken ist aber, dass es ungleich dem Sport keine "ordentliche" Zielmarkierung gibt, der Fahrgast wird sich mit etwas wie "aus dem Augenwinkel verschwunden" behelfen muessen, falls er nicht zufaellig eine Kerbe oder dergleichen im Fenster hat. Anmerkung: Keine Kerben zur Bestimmung der Fahrtgeschwindigkeit in Bahnfenster ritzen.
Also, welche Ungenauigkeit bringt ein Messfehler von plusminus 0,3s?
Was da auf den ersten Blick doch eigentlich ganz huebsch aussieht, wird zimlich verheerend, wenn man sich mal den Bereich zwischen 100 und 250 km/h anschaut:
Angenommen, es werden 4 Sekunden mit einer Messungenauigkeit von 0,3s gemessen. Dann bedeutet das nichts anderes, als dass sich die Zuggeschwindigkeit irgendwo zwischen 166 und 195 km/h bewegt. Das ist -- freundlich ausgedrueckt -- unschoen. Aber wie in der vorherigen Graphik zu sehen wird der Einfluss eines konstanten Messfehlers (welchen ich hier einfach einmal anzunehmen wage) hin zu groesseren Messwerten geringer. Also: Laengere Messung, besserer Geschwindigkeitsschaetzer. Aber (ohne an Heisenberg zu denken), man kann nicht alles haben. Zwischen zwei Hektometertafeln ermittelt man mit dem vorgestellten Verfahren die Durchschnittsgeschwindigkeit ueber 200m. Wuerde man eine Hektometertafel spaeter stoppen (also 400m betrachten), bezieht sich der Ergebniswert auch auf diese groessere Strecke. Also, nochmal geplottet fuer 400m Beobachtungsstrecke.
Bei 8s Messzeit ergeben sich also idealerweise 180km/h -- mit einer Messungenauigkeit von 0,3s kann es alles zwischen 173 und 187km/h sein. Sind immer noch knapp 20km/h Messfehler. Spasseshalber also nochmal fuer 600m (3 Hektometertafeln):
Bei 12s Messzeit ergeben sich also 180 km/h (bzw. mit Ungenauigkeit irgendwas zwischen 175 und 185 km/h). Langsam wird's ertraeglich und nun soll's auch mit der Plotterei reichen. Wenn man einen neuen Hammer gefunden hat und so :)

Es sei aber angemerkt, dass ein Messfehler von nur 0,3 eine zimlich optimistische Annahme fuer die Bedingungen in einem Zug ist. Als weitere Faktoren kommen in die Ungenauigkeit natuerlich noch Winkelverschiebungen durch unterschiedlich weit vom Zug entfernt stehende Hektometertafeln hinzu...

Weitergedacht

Natuerlich eignet sich diese Idee wunderbar, um eine App fuers Smartphone daraus zu entwickeln. Mein Vorschlag: Ein grosser Button fuer die Stoppuhrbedienung (Start-Stopp) und vier Ausgabefelder fuer die jeweilige Durchschnittsgeschwindigkeit fuer den Fall, dass n=1,..,4 vorbeiziehende Hektometertafeln betrachtet wurden. Schoen waere auch die Berechnung der Ungenauigkeit fuer eine vom Nutzer vorgegebene Ungenauigkeit beim Stoppen der Messzeit.

Tabellen

Hier also nun die Tabellen zur Geschwindigkeitsermitttlung ueber unterschiedlich viele Hektometertafeln. Die gibt es auch als PDF zum Ausdrucken -- down/bahngeschwindigkeiten.pdf.

t n=1 n=2 n=3 n=4
0.17200.014400.021600.028800
0.23600.07200.010800.014400
0.32400.04800.07200.09600
0.41800.03600.05400.07200
0.51440.02880.04320.05760
0.61200.02400.03600.04800
0.71028.62057.13085.74114
0.8900.01800.02700.03600
0.9800.01600.02400.03200
1.0720.01440.02160.02880
1.1654.51309.11963.62618
1.2600.01200.01800.02400
1.3553.81107.71661.52215
1.4514.31028.61542.92057
1.5480.0960.01440.01920
1.6450.0900.01350.01800
1.7423.5847.11270.61694
1.8400.0800.01200.01600
1.9378.9757.91136.81516
2.0360.0720.01080.01440
2.1342.9685.71028.61371
2.2327.3654.5981.81309
2.3313.0626.1939.11252
2.4300.0600.0900.01200
2.5288.0576.0864.01152
2.6276.9553.8830.81108
2.7266.7533.3800.01067
2.8257.1514.3771.41029
2.9248.3496.6744.8993
3.0240.0480.0720.0960
3.1232.3464.5696.8929
3.2225.0450.0675.0900
3.3218.2436.4654.5873
3.4211.8423.5635.3847
3.5205.7411.4617.1823
3.6200.0400.0600.0800
3.7194.6389.2583.8778
3.8189.5378.9568.4758
3.9184.6369.2553.8738
4.0180.0360.0540.0720
4.1175.6351.2526.8702
4.2171.4342.9514.3686
4.3167.4334.9502.3670
4.4163.6327.3490.9655
4.5160.0320.0480.0640
4.6156.5313.0469.6626
4.7153.2306.4459.6613
4.8150.0300.0450.0600
4.9146.9293.9440.8588
5.0144.0288.0432.0576
5.1141.2282.4423.5565
5.2138.5276.9415.4554
5.3135.8271.7407.5543
5.4133.3266.7400.0533
5.5130.9261.8392.7524
5.6128.6257.1385.7514
5.7126.3252.6378.9505
5.8124.1248.3372.4497
5.9122.0244.1366.1488
6.0120.0240.0360.0480
6.1118.0236.1354.1472
6.2116.1232.3348.4465
6.3114.3228.6342.9457
6.4112.5225.0337.5450
6.5110.8221.5332.3443
6.6109.1218.2327.3436
6.7107.5214.9322.4430
6.8105.9211.8317.6424
6.9104.3208.7313.0417
7.0102.9205.7308.6411
7.1101.4202.8304.2406
7.2100.0200.0300.0400
7.398.6197.3295.9395
7.497.3194.6291.9389
7.596.0192.0288.0384
7.694.7189.5284.2379
7.793.5187.0280.5374
7.892.3184.6276.9369
7.991.1182.3273.4365
8.090.0180.0270.0360
8.188.9177.8266.7356
8.287.8175.6263.4351
8.386.7173.5260.2347
8.485.7171.4257.1343
8.584.7169.4254.1339
8.683.7167.4251.2335
8.782.8165.5248.3331
8.881.8163.6245.5327
8.980.9161.8242.7324
9.080.0160.0240.0320
9.179.1158.2237.4316
9.278.3156.5234.8313
9.377.4154.8232.3310
9.476.6153.2229.8306
9.575.8151.6227.4303
9.675.0150.0225.0300
9.774.2148.5222.7297
9.873.5146.9220.4294
9.972.7145.5218.2291
10.072.0144.0216.0288
10.171.3142.6213.9285
10.270.6141.2211.8282
10.369.9139.8209.7280
10.469.2138.5207.7277
10.568.6137.1205.7274
10.667.9135.8203.8272
10.767.3134.6201.9269
10.866.7133.3200.0267
10.966.1132.1198.2264
11.065.5130.9196.4262
11.164.9129.7194.6259
11.264.3128.6192.9257
11.363.7127.4191.2255
11.463.2126.3189.5253
11.562.6125.2187.8250
11.662.1124.1186.2248
11.761.5123.1184.6246
11.861.0122.0183.1244
11.960.5121.0181.5242
12.060.0120.0180.0240
12.159.5119.0178.5238
12.259.0118.0177.0236
12.358.5117.1175.6234
12.458.1116.1174.2232
12.557.6115.2172.8230
12.657.1114.3171.4229
12.756.7113.4170.1227
12.856.2112.5168.8225
12.955.8111.6167.4223
13.055.4110.8166.2222
13.155.0109.9164.9220
13.254.5109.1163.6218
13.354.1108.3162.4217
13.453.7107.5161.2215
13.553.3106.7160.0213
13.652.9105.9158.8212
13.752.6105.1157.7210
13.852.2104.3156.5209
13.951.8103.6155.4207
14.051.4102.9154.3206
14.151.1102.1153.2204
14.250.7101.4152.1203
14.350.3100.7151.0201
14.450.0100.0150.0200
14.549.799.3149.0199
14.649.398.6147.9197
14.749.098.0146.9196
14.848.697.3145.9195
14.948.396.6145.0193
15.048.096.0144.0192
15.147.795.4143.0191
15.247.494.7142.1189
15.347.194.1141.2188
15.446.893.5140.3187
15.546.592.9139.4186
15.646.292.3138.5185
15.745.991.7137.6183
15.845.691.1136.7182
15.945.390.6135.8181
16.045.090.0135.0180
16.144.789.4134.2179
16.244.488.9133.3178
16.344.288.3132.5177
16.443.987.8131.7176
16.543.687.3130.9175
16.643.486.7130.1173
16.743.186.2129.3172
16.842.985.7128.6171
16.942.685.2127.8170
17.042.484.7127.1169
17.142.184.2126.3168
17.241.983.7125.6167
17.341.683.2124.9166
17.441.482.8124.1166
17.541.182.3123.4165
17.640.981.8122.7164
17.740.781.4122.0163
17.840.480.9121.3162
17.940.280.4120.7161
18.040.080.0120.0160
18.139.879.6119.3159
18.239.679.1118.7158
18.339.378.7118.0157
18.439.178.3117.4157
18.538.977.8116.8156
18.638.777.4116.1155
18.738.577.0115.5154
18.838.376.6114.9153
18.938.176.2114.3152
19.037.975.8113.7152
19.137.775.4113.1151
19.237.575.0112.5150
19.337.374.6111.9149
19.437.174.2111.3148
19.536.973.8110.8148
19.636.773.5110.2147
19.736.573.1109.6146
19.836.472.7109.1145
19.936.272.4108.5145
20.036.072.0108.0144
20.135.871.6107.5143
20.235.671.3106.9143
20.335.570.9106.4142
20.435.370.6105.9141
20.535.170.2105.4140
20.635.069.9104.9140
20.734.869.6104.3139
20.834.669.2103.8138
20.934.468.9103.3138
21.034.368.6102.9137
21.134.168.2102.4136
21.234.067.9101.9136
21.333.867.6101.4135
21.433.667.3100.9135
21.533.567.0100.5134
21.633.366.7100.0133
21.733.266.499.5133
21.833.066.199.1132
21.932.965.898.6132
22.032.765.598.2131
22.132.665.297.7130
22.232.464.997.3130
22.332.364.696.9129
22.432.164.396.4129
22.532.064.096.0128
22.631.963.795.6127
22.731.763.495.2127
22.831.663.294.7126
22.931.462.994.3126
23.031.362.693.9125
23.131.262.393.5125
23.231.062.193.1124
23.330.961.892.7124
23.430.861.592.3123
23.530.661.391.9123
23.630.561.091.5122
23.730.460.891.1122
23.830.360.590.8121
23.930.160.390.4121
24.030.060.090.0120
24.129.959.889.6120
24.229.859.589.3119
24.329.659.388.9119
24.429.559.088.5118
24.529.458.888.2118
24.629.358.587.8117
24.729.158.387.4117
24.829.058.187.1116
24.928.957.886.7116
25.028.857.686.4115
25.128.757.486.1115
25.228.657.185.7114
25.328.556.985.4114
25.428.356.785.0113
25.528.256.584.7113
25.628.156.284.4112
25.728.056.084.0112
25.827.955.883.7112
25.927.855.683.4111
26.027.755.483.1111
26.127.655.282.8110
26.227.555.082.4110
26.327.454.882.1110
26.427.354.581.8109
26.527.254.381.5109
26.627.154.181.2108
26.727.053.980.9108
26.826.953.780.6107
26.926.853.580.3107
27.026.753.380.0107
27.126.653.179.7106
27.226.552.979.4106
27.326.452.779.1105
27.426.352.678.8105
27.526.252.478.5105
27.626.152.278.3104
27.726.052.078.0104
27.825.951.877.7104
27.925.851.677.4103
28.025.751.477.1103
28.125.651.276.9102
28.225.551.176.6102
28.325.450.976.3102
28.425.450.776.1101
28.525.350.575.8101
28.625.250.375.5101
28.725.150.275.3100
28.825.050.075.0100
28.924.949.874.7100
29.024.849.774.599
29.124.749.574.299
29.224.749.374.099
29.324.649.173.798
29.424.549.073.598
29.524.448.873.298
29.624.348.673.097
29.724.248.572.797
29.824.248.372.597
29.924.148.272.296
30.024.048.072.096
30.123.947.871.896
30.223.847.771.595
30.323.847.571.395
30.423.747.471.195
30.523.647.270.894
30.623.547.170.694
30.723.546.970.494
30.823.446.870.194
30.923.346.669.993
31.023.246.569.793
31.123.246.369.593
31.223.146.269.292
31.323.046.069.092
31.422.945.968.892
31.522.945.768.691
31.622.845.668.491
31.722.745.468.191
31.822.645.367.991
31.922.645.167.790
32.022.545.067.590
32.122.444.967.390
32.222.444.767.189
32.322.344.666.989
32.422.244.466.789
32.522.244.366.589
32.622.144.266.388
32.722.044.066.188
32.822.043.965.988
32.921.943.865.788
33.021.843.665.587
33.121.843.565.387
33.221.743.465.187
33.321.643.264.986
33.421.643.164.786
33.521.543.064.586
33.621.442.964.386
33.721.442.764.185
33.821.342.663.985
33.921.242.563.785
34.021.242.463.585
34.121.142.263.384
34.221.142.163.284
34.321.042.063.084
34.420.941.962.884
34.520.941.762.683
34.620.841.662.483
34.720.741.562.283
34.820.741.462.183
34.920.641.361.983
35.020.641.161.782
35.120.541.061.582
35.220.540.961.482
35.320.440.861.282
35.420.340.761.081
35.520.340.660.881
35.620.240.460.781
35.720.240.360.581
35.820.140.260.380
35.920.140.160.280
36.020.040.060.080
36.119.939.959.880
36.219.939.859.780
36.319.839.759.579
36.419.839.659.379
36.519.739.559.279
36.619.739.359.079
36.719.639.258.978
36.819.639.158.778
36.919.539.058.578
37.019.538.958.478
37.119.438.858.278
37.219.438.758.177
37.319.338.657.977
37.419.338.557.877
37.519.238.457.677
37.619.138.357.477
37.719.138.257.376
37.819.038.157.176
37.919.038.057.076
38.018.937.956.876
38.118.937.856.776
38.218.837.756.575
38.318.837.656.475
38.418.837.556.275
38.518.737.456.175
38.618.737.356.075
38.718.637.255.874
38.818.637.155.774
38.918.537.055.574
39.018.536.955.474
39.118.436.855.274
39.218.436.755.173
39.318.336.655.073
39.418.336.554.873
39.518.236.554.773
39.618.236.454.573
39.718.136.354.473
39.818.136.254.372
39.918.036.154.172
40.018.036.054.072
40.118.035.953.972
40.217.935.853.772
40.317.935.753.671
40.417.835.653.571
40.517.835.653.371
40.617.735.553.271
40.717.735.453.171
40.817.635.352.971
40.917.635.252.870
41.017.635.152.770
41.117.535.052.670
41.217.535.052.470
41.317.434.952.370
41.417.434.852.270
41.517.334.752.069
41.617.334.651.969
41.717.334.551.869
41.817.234.451.769
41.917.234.451.669
42.017.134.351.469
42.117.134.251.368
42.217.134.151.268
42.317.034.051.168
42.417.034.050.968
42.516.933.950.868
42.616.933.850.768
42.716.933.750.667
42.816.833.650.567
42.916.833.650.367
43.016.733.550.267
43.116.733.450.167
43.216.733.350.067
43.316.633.349.967
43.416.633.249.866
43.516.633.149.766
43.616.533.049.566
43.716.533.049.466
43.816.432.949.366
43.916.432.849.266
44.016.432.749.165
44.116.332.749.065
44.216.332.648.965
44.316.332.548.865
44.416.232.448.665
44.516.232.448.565
44.616.132.348.465
44.716.132.248.364
44.816.132.148.264
44.916.032.148.164
45.016.032.048.064
45.116.031.947.964
45.215.931.947.864
45.315.931.847.764
45.415.931.747.663
45.515.831.647.563
45.615.831.647.463
45.715.831.547.363
45.815.731.447.263
45.915.731.447.163
46.015.731.347.063
46.115.631.246.962
46.215.631.246.862
46.315.631.146.762
46.415.531.046.662
46.515.531.046.562
46.615.530.946.462
46.715.430.846.362
46.815.430.846.262
46.915.430.746.161
47.015.330.646.061
47.115.330.645.961
47.215.330.545.861
47.315.230.445.761
47.415.230.445.661
47.515.230.345.561
47.615.130.345.461
47.715.130.245.360
47.815.130.145.260
47.915.030.145.160
48.015.030.045.060
48.115.029.944.960
48.214.929.944.860
48.314.929.844.760
48.414.929.844.660
48.514.829.744.559
48.614.829.644.459
48.714.829.644.459
48.814.829.544.359
48.914.729.444.259
49.014.729.444.159
49.114.729.344.059
49.214.629.343.959
49.314.629.243.858
49.414.629.143.758
49.514.529.143.658
49.614.529.043.558
49.714.529.043.558
49.814.528.943.458
49.914.428.943.358
50.014.428.843.258
50.114.428.743.157
50.214.328.743.057
50.314.328.642.957
50.414.328.642.957
50.514.328.542.857
50.614.228.542.757
50.714.228.442.657
50.814.228.342.557
50.914.128.342.457
51.014.128.242.456
51.114.128.242.356
51.214.128.142.256
51.314.028.142.156
51.414.028.042.056
51.514.028.041.956
51.614.027.941.956
51.713.927.941.856
51.813.927.841.756
51.913.927.741.655
52.013.827.741.555
52.113.827.641.555
52.213.827.641.455
52.313.827.541.355
52.413.727.541.255
52.513.727.441.155
52.613.727.441.155
52.713.727.341.055
52.813.627.340.955
52.913.627.240.854
53.013.627.240.854
53.113.627.140.754
53.213.527.140.654
53.313.527.040.554
53.413.527.040.454
53.513.526.940.454
53.613.426.940.354
53.713.426.840.254
53.813.426.840.154
53.913.426.740.153
54.013.326.740.053
54.113.326.639.953
54.213.326.639.953
54.313.326.539.853
54.413.226.539.753
54.513.226.439.653
54.613.226.439.653
54.713.226.339.553
54.813.126.339.453
54.913.126.239.352
55.013.126.239.352
55.113.126.139.252
55.213.026.139.152
55.313.026.039.152
55.413.026.039.052
55.513.025.938.952
55.612.925.938.852
55.712.925.938.852
55.812.925.838.752
55.912.925.838.652
56.012.925.738.651
56.112.825.738.551
56.212.825.638.451
56.312.825.638.451
56.412.825.538.351
56.512.725.538.251
56.612.725.438.251
56.712.725.438.151
56.812.725.438.051
56.912.725.338.051
57.012.625.337.951
57.112.625.237.850
57.212.625.237.850
57.312.625.137.750
57.412.525.137.650
57.512.525.037.650
57.612.525.037.550
57.712.525.037.450
57.812.524.937.450
57.912.424.937.350
58.012.424.837.250
58.112.424.837.250
58.212.424.737.149
58.312.324.737.049
58.412.324.737.049
58.512.324.636.949
58.612.324.636.949
58.712.324.536.849
58.812.224.536.749
58.912.224.436.749
59.012.224.436.649
59.112.224.436.549
59.212.224.336.549
59.312.124.336.449
59.412.124.236.448
59.512.124.236.348
59.612.124.236.248
59.712.124.136.248
59.812.024.136.148
59.912.024.036.148
60.012.024.036.048
60.112.024.035.948
60.212.023.935.948
60.311.923.935.848
60.411.923.835.848
60.511.923.835.748
60.611.923.835.648
60.711.923.735.647
60.811.823.735.547
60.911.823.635.547
61.011.823.635.447
61.111.823.635.447
61.211.823.535.347
61.311.723.535.247
61.411.723.535.247
61.511.723.435.147
61.611.723.435.147
61.711.723.335.047
61.811.723.335.047
61.911.623.334.947
62.011.623.234.846
62.111.623.234.846
62.211.623.234.746
62.311.623.134.746
62.411.523.134.646
62.511.523.034.646
62.611.523.034.546
62.711.523.034.446
62.811.522.934.446
62.911.422.934.346
63.011.422.934.346
63.111.422.834.246
63.211.422.834.246
63.311.422.734.145
63.411.422.734.145
63.511.322.734.045
63.611.322.634.045
63.711.322.633.945
63.811.322.633.945
63.911.322.533.845
64.011.222.533.845
64.111.222.533.745
64.211.222.433.645
64.311.222.433.645
64.411.222.433.545
64.511.222.333.545
64.611.122.333.445
64.711.122.333.445
64.811.122.233.344
64.911.122.233.344
65.011.122.233.244
65.111.122.133.244
65.211.022.133.144
65.311.022.133.144
65.411.022.033.044
65.511.022.033.044
65.611.022.032.944
65.711.021.932.944
65.810.921.932.844
65.910.921.932.844
66.010.921.832.744
66.110.921.832.744
66.210.921.832.644
66.310.921.732.643
66.410.821.732.543
66.510.821.732.543
66.610.821.632.443
66.710.821.632.443
66.810.821.632.343
66.910.821.532.343
67.010.721.532.243
67.110.721.532.243
67.210.721.432.143
67.310.721.432.143
67.410.721.432.043
67.510.721.332.043
67.610.721.332.043
67.710.621.331.943
67.810.621.231.942
67.910.621.231.842
68.010.621.231.842
68.110.621.131.742
68.210.621.131.742
68.310.521.131.642
68.410.521.131.642
68.510.521.031.542
68.610.521.031.542
68.710.521.031.442
68.810.520.931.442
68.910.420.931.342
69.010.420.931.342
69.110.420.831.342
69.210.420.831.242
69.310.420.831.242
69.410.420.731.141
69.510.420.731.141
69.610.320.731.041
69.710.320.731.041
69.810.320.630.941
69.910.320.630.941
70.010.320.630.941
70.110.320.530.841
70.210.320.530.841
70.310.220.530.741
70.410.220.530.741
70.510.220.430.641
70.610.220.430.641
70.710.220.430.641
70.810.220.330.541
70.910.220.330.541
71.010.120.330.441
71.110.120.330.441
71.210.120.230.340
71.310.120.230.340
71.410.120.230.340
71.510.120.130.240
71.610.120.130.240
71.710.020.130.140
71.810.020.130.140
71.910.020.030.040
72.010.020.030.040
72.110.020.030.040
72.210.019.929.940
72.310.019.929.940
72.49.919.929.840
72.59.919.929.840
72.69.919.829.840
72.79.919.829.740
72.89.919.829.740
72.99.919.829.640
73.09.919.729.639
73.19.819.729.539
73.29.819.729.539
73.39.819.629.539
73.49.819.629.439
73.59.819.629.439
73.69.819.629.339
73.79.819.529.339
73.89.819.529.339
73.99.719.529.239
74.09.719.529.239
74.19.719.429.139
74.29.719.429.139
74.39.719.429.139
74.49.719.429.039
74.59.719.329.039
74.69.719.329.039
74.79.619.328.939
74.89.619.328.939
74.99.619.228.838
75.09.619.228.838
75.19.619.228.838
75.29.619.128.738
75.39.619.128.738
75.49.519.128.638
75.59.519.128.638
75.69.519.028.638
75.79.519.028.538
75.89.519.028.538
75.99.519.028.538
76.09.518.928.438
76.19.518.928.438
76.29.418.928.338
76.39.418.928.338
76.49.418.828.338
76.59.418.828.238
76.69.418.828.238
76.79.418.828.238
76.89.418.828.138
76.99.418.728.137
77.09.418.728.137
77.19.318.728.037
77.29.318.728.037
77.39.318.627.937
77.49.318.627.937
77.59.318.627.937
77.69.318.627.837
77.79.318.527.837
77.89.318.527.837
77.99.218.527.737
78.09.218.527.737
78.19.218.427.737
78.29.218.427.637
78.39.218.427.637
78.49.218.427.637
78.59.218.327.537
78.69.218.327.537
78.79.118.327.437
78.89.118.327.437
78.99.118.327.437
79.09.118.227.336
79.19.118.227.336
79.29.118.227.336
79.39.118.227.236
79.49.118.127.236
79.59.118.127.236
79.69.018.127.136
79.79.018.127.136
79.89.018.027.136
79.99.018.027.036
80.09.018.027.036
80.19.018.027.036
80.29.018.026.936
80.39.017.926.936
80.49.017.926.936
80.58.917.926.836
80.68.917.926.836
80.78.917.826.836
80.88.917.826.736
80.98.917.826.736
81.08.917.826.736
81.18.917.826.636
81.28.917.726.635
81.38.917.726.635
81.48.817.726.535
81.58.817.726.535
81.68.817.626.535
81.78.817.626.435
81.88.817.626.435
81.98.817.626.435
82.08.817.626.335
82.18.817.526.335
82.28.817.526.335
82.38.717.526.235
82.48.717.526.235
82.58.717.526.235
82.68.717.426.235
82.78.717.426.135
82.88.717.426.135
82.98.717.426.135
83.08.717.326.035
83.18.717.326.035
83.28.717.326.035
83.38.617.325.935
83.48.617.325.935
83.58.617.225.934
83.68.617.225.834
83.78.617.225.834
83.88.617.225.834
83.98.617.225.734
84.08.617.125.734
84.18.617.125.734
84.28.617.125.734
84.38.517.125.634
84.48.517.125.634
84.58.517.025.634
84.68.517.025.534
84.78.517.025.534
84.88.517.025.534
84.98.517.025.434
85.08.516.925.434
85.18.516.925.434
85.28.516.925.434
85.38.416.925.334
85.48.416.925.334
85.58.416.825.334
85.68.416.825.234
85.78.416.825.234
85.88.416.825.234
85.98.416.825.134
86.08.416.725.133
86.18.416.725.133
86.28.416.725.133
86.38.316.725.033
86.48.316.725.033
86.58.316.625.033
86.68.316.624.933
86.78.316.624.933
86.88.316.624.933
86.98.316.624.933
87.08.316.624.833
87.18.316.524.833
87.28.316.524.833
87.38.216.524.733
87.48.216.524.733
87.58.216.524.733
87.68.216.424.733
87.78.216.424.633
87.88.216.424.633
87.98.216.424.633
88.08.216.424.533
88.18.216.324.533
88.28.216.324.533
88.38.216.324.533
88.48.116.324.433
88.58.116.324.433
88.68.116.324.433
88.78.116.224.432
88.88.116.224.332
88.98.116.224.332
89.08.116.224.332
89.18.116.224.232
89.28.116.124.232
89.38.116.124.232
89.48.116.124.232
89.58.016.124.132
89.68.016.124.132
89.78.016.124.132
89.88.016.024.132
89.98.016.024.032
90.08.016.024.032
90.18.016.024.032
90.28.016.023.932
90.38.015.923.932
90.48.015.923.932
90.58.015.923.932
90.67.915.923.832
90.77.915.923.832
90.87.915.923.832
90.97.915.823.832
91.07.915.823.732
91.17.915.823.732
91.27.915.823.732
91.37.915.823.732
91.47.915.823.632
91.57.915.723.631
91.67.915.723.631
91.77.915.723.631
91.87.815.723.531
91.97.815.723.531
92.07.815.723.531
92.17.815.623.531
92.27.815.623.431
92.37.815.623.431
92.47.815.623.431
92.57.815.623.431
92.67.815.623.331
92.77.815.523.331
92.87.815.523.331
92.97.815.523.331
93.07.715.523.231
93.17.715.523.231
93.27.715.523.231
93.37.715.423.231
93.47.715.423.131
93.57.715.423.131
93.67.715.423.131
93.77.715.423.131
93.87.715.423.031
93.97.715.323.031
94.07.715.323.031
94.17.715.323.031
94.27.615.322.931
94.37.615.322.931
94.47.615.322.931
94.57.615.222.930
94.67.615.222.830
94.77.615.222.830
94.87.615.222.830
94.97.615.222.830
95.07.615.222.730
95.17.615.122.730
95.27.615.122.730
95.37.615.122.730
95.47.515.122.630
95.57.515.122.630
95.67.515.122.630
95.77.515.022.630
95.87.515.022.530
95.97.515.022.530
96.07.515.022.530
96.17.515.022.530
96.27.515.022.530
96.37.515.022.430
96.47.514.922.430
96.57.514.922.430
96.67.514.922.430
96.77.414.922.330
96.87.414.922.330
96.97.414.922.330
97.07.414.822.330
97.17.414.822.230
97.27.414.822.230
97.37.414.822.230
97.47.414.822.230
97.57.414.822.230
97.67.414.822.130
97.77.414.722.129
97.87.414.722.129
97.97.414.722.129
98.07.314.722.029
98.17.314.722.029
98.27.314.722.029
98.37.314.622.029
98.47.314.622.029
98.57.314.621.929
98.67.314.621.929
98.77.314.621.929
98.87.314.621.929
98.97.314.621.829
99.07.314.521.829
99.17.314.521.829
99.27.314.521.829
99.37.314.521.829
99.47.214.521.729
99.57.214.521.729
99.67.214.521.729
99.77.214.421.729
99.87.214.421.629
99.97.214.421.629

Stichworte:


Impressum