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An Amazing Relationship?
Problem 1:
Suppose you are riding a motorcycle at the speed of 40 mph and about to enter a level, circular, right-hander curve whose radius is 100 yards. You intend to lean with the bike and want to make negotiating the curve as smooth and effortless as possible. What would be the lean angle of the bike during the turn?
Problem 2:
Suppose you are riding a motorcycle whose maximum right-side lean angle attainable without dragging any part of it is 30 degrees. You are doing 80 mph and approaching a level, circular, right-hander curve whose radius is 100 yards. You intend to lean with the bike and want to make negotiating the curve as smooth and effortless as possible. What would be the entry speed you have to reduce the current speed to, to safely negotiate the curve without dragging any part of the bike?
Strange as it may seem, you can obtain approximated answers to these kinds of problems from the kind of information given in the above problems using only one relationship.
The following table and chart were prepared using the relationship:
Table: Lean Angles by Speed and Turn Radius
| Speed | Turn Radius (Yards) | |||||||||||||||||||||||||||||||||||||||
| (mph) | 5 | 10 | 15 | 20 | 25 | 30 | 35 | 40 | 45 | 50 | 55 | 60 | 65 | 70 | 75 | 80 | 85 | 90 | 95 | 100 | 105 | 110 | 115 | 120 | 125 | 130 | 135 | 140 | 145 | 150 | 155 | 160 | 165 | 170 | 175 | 180 | 185 | 190 | 195 | 200 |
| 5 | 6.4 | 3.2 | 2.1 | 1.6 | 1.3 | 1.1 | 0.9 | 0.8 | 0.7 | 0.6 | 0.6 | 0.5 | 0.5 | 0.5 | 0.4 | 0.4 | 0.4 | 0.4 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.3 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 | 0.2 |
| 10 | 24.0 | 12.6 | 8.5 | 6.4 | 5.1 | 4.3 | 3.6 | 3.2 | 2.8 | 2.6 | 2.3 | 2.1 | 2.0 | 1.8 | 1.7 | 1.6 | 1.5 | 1.4 | 1.3 | 1.3 | 1.2 | 1.2 | 1.1 | 1.1 | 1.0 | 1.0 | 0.9 | 0.9 | 0.9 | 0.9 | 0.8 | 0.8 | 0.8 | 0.8 | 0.7 | 0.7 | 0.7 | 0.7 | 0.7 | 0.6 |
| 15 | 45.1 | 26.6 | 18.5 | 14.1 | 11.3 | 9.5 | 8.2 | 7.2 | 6.4 | 5.7 | 5.2 | 4.8 | 4.4 | 4.1 | 3.8 | 3.6 | 3.4 | 3.2 | 3.0 | 2.9 | 2.7 | 2.6 | 2.5 | 2.4 | 2.3 | 2.2 | 2.1 | 2.1 | 2.0 | 1.9 | 1.9 | 1.8 | 1.7 | 1.7 | 1.6 | 1.6 | 1.6 | 1.5 | 1.5 | 1.4 |
| 20 | 60.7 | 41.7 | 30.7 | 24.0 | 19.6 | 16.6 | 14.3 | 12.6 | 11.2 | 10.1 | 9.2 | 8.5 | 7.8 | 7.3 | 6.8 | 6.4 | 6.0 | 5.7 | 5.4 | 5.1 | 4.9 | 4.6 | 4.4 | 4.3 | 4.1 | 3.9 | 3.8 | 3.6 | 3.5 | 3.4 | 3.3 | 3.2 | 3.1 | 3.0 | 2.9 | 2.8 | 2.8 | 2.7 | 2.6 | 2.6 |
| 25 | 70.3 | 54.3 | 42.9 | 34.9 | 29.1 | 24.9 | 21.7 | 19.2 | 17.2 | 15.6 | 14.2 | 13.1 | 12.1 | 11.3 | 10.5 | 9.9 | 9.3 | 8.8 | 8.3 | 7.9 | 7.6 | 7.2 | 6.9 | 6.6 | 6.4 | 6.1 | 5.9 | 5.7 | 5.5 | 5.3 | 5.1 | 5.0 | 4.8 | 4.7 | 4.6 | 4.4 | 4.3 | 4.2 | 4.1 | 4.0 |
| 30 | 76.0 | 63.5 | 53.2 | 45.1 | 38.8 | 33.8 | 29.8 | 26.6 | 24.0 | 21.9 | 20.0 | 18.5 | 17.2 | 16.0 | 15.0 | 14.1 | 13.3 | 12.6 | 11.9 | 11.3 | 10.8 | 10.3 | 9.9 | 9.5 | 9.1 | 8.8 | 8.5 | 8.2 | 7.9 | 7.6 | 7.4 | 7.2 | 6.9 | 6.7 | 6.5 | 6.4 | 6.2 | 6.0 | 5.9 | 5.7 |
| 35 | 79.6 | 69.9 | 61.2 | 53.8 | 47.5 | 42.3 | 38.0 | 34.3 | 31.3 | 28.7 | 26.4 | 24.5 | 22.8 | 21.3 | 20.0 | 18.9 | 17.8 | 16.9 | 16.0 | 15.3 | 14.6 | 13.9 | 13.4 | 12.8 | 12.3 | 11.9 | 11.4 | 11.0 | 10.7 | 10.3 | 10.0 | 9.7 | 9.4 | 9.1 | 8.9 | 8.6 | 8.4 | 8.2 | 8.0 | 7.8 |
| 40 | 82.0 | 74.3 | 67.2 | 60.7 | 55.0 | 49.9 | 45.6 | 41.7 | 38.4 | 35.5 | 33.0 | 30.7 | 28.8 | 27.0 | 25.4 | 24.0 | 22.8 | 21.6 | 20.6 | 19.6 | 18.8 | 18.0 | 17.2 | 16.6 | 15.9 | 15.3 | 14.8 | 14.3 | 13.8 | 13.4 | 13.0 | 12.6 | 12.2 | 11.9 | 11.5 | 11.2 | 10.9 | 10.6 | 10.4 | 10.1 |
| 45 | 83.7 | 77.5 | 71.6 | 66.1 | 61.0 | 56.4 | 52.2 | 48.5 | 45.1 | 42.1 | 39.4 | 37.0 | 34.8 | 32.8 | 31.1 | 29.4 | 28.0 | 26.6 | 25.4 | 24.3 | 23.3 | 22.3 | 21.4 | 20.6 | 19.9 | 19.2 | 18.5 | 17.9 | 17.3 | 16.8 | 16.2 | 15.8 | 15.3 | 14.9 | 14.5 | 14.1 | 13.7 | 13.4 | 13.0 | 12.7 |
| 50 | 84.9 | 79.8 | 74.9 | 70.3 | 65.8 | 61.7 | 57.9 | 54.3 | 51.1 | 48.1 | 45.4 | 42.9 | 40.6 | 38.5 | 36.6 | 34.9 | 33.3 | 31.8 | 30.4 | 29.1 | 28.0 | 26.9 | 25.9 | 24.9 | 24.0 | 23.2 | 22.4 | 21.7 | 21.0 | 20.4 | 19.8 | 19.2 | 18.7 | 18.2 | 17.7 | 17.2 | 16.8 | 16.4 | 16.0 | 15.6 |
| 55 | 85.8 | 81.6 | 77.5 | 73.5 | 69.7 | 66.0 | 62.6 | 59.3 | 56.3 | 53.5 | 50.8 | 48.4 | 46.1 | 43.9 | 42.0 | 40.1 | 38.4 | 36.9 | 35.4 | 34.0 | 32.7 | 31.5 | 30.4 | 29.3 | 28.4 | 27.4 | 26.6 | 25.7 | 25.0 | 24.2 | 23.5 | 22.9 | 22.2 | 21.6 | 21.1 | 20.5 | 20.0 | 19.5 | 19.1 | 18.6 |
| 60 | 86.4 | 82.9 | 79.4 | 76.0 | 72.7 | 69.5 | 66.4 | 63.5 | 60.7 | 58.1 | 55.6 | 53.2 | 51.0 | 48.9 | 47.0 | 45.1 | 43.4 | 41.7 | 40.2 | 38.8 | 37.4 | 36.1 | 34.9 | 33.8 | 32.7 | 31.7 | 30.7 | 29.8 | 29.0 | 28.2 | 27.4 | 26.6 | 25.9 | 25.3 | 24.6 | 24.0 | 23.5 | 22.9 | 22.4 | 21.9 |
| 65 | 87.0 | 83.9 | 81.0 | 78.0 | 75.1 | 72.3 | 69.6 | 67.0 | 64.5 | 62.0 | 59.7 | 57.5 | 55.4 | 53.4 | 51.5 | 49.7 | 47.9 | 46.3 | 44.8 | 43.3 | 41.9 | 40.6 | 39.3 | 38.1 | 37.0 | 35.9 | 34.9 | 33.9 | 33.0 | 32.1 | 31.3 | 30.5 | 29.7 | 29.0 | 28.3 | 27.6 | 27.0 | 26.4 | 25.8 | 25.2 |
| 70 | 87.4 | 84.8 | 82.2 | 79.6 | 77.1 | 74.6 | 72.2 | 69.9 | 67.6 | 65.4 | 63.3 | 61.2 | 59.3 | 57.4 | 55.5 | 53.8 | 52.1 | 50.5 | 49.0 | 47.5 | 46.1 | 44.8 | 43.5 | 42.3 | 41.2 | 40.1 | 39.0 | 38.0 | 37.0 | 36.1 | 35.2 | 34.3 | 33.5 | 32.7 | 32.0 | 31.3 | 30.6 | 29.9 | 29.3 | 28.7 |
| 75 | 87.7 | 85.4 | 83.2 | 80.9 | 78.7 | 76.6 | 74.4 | 72.3 | 70.3 | 68.3 | 66.3 | 64.4 | 62.6 | 60.8 | 59.1 | 57.5 | 55.9 | 54.3 | 52.9 | 51.4 | 50.1 | 48.8 | 47.5 | 46.3 | 45.1 | 44.0 | 42.9 | 41.9 | 40.9 | 39.9 | 39.0 | 38.1 | 37.2 | 36.4 | 35.6 | 34.9 | 34.1 | 33.4 | 32.8 | 32.1 |
| 80 | 88.0 | 86.0 | 84.0 | 82.0 | 80.1 | 78.1 | 76.2 | 74.3 | 72.5 | 70.7 | 68.9 | 67.2 | 65.5 | 63.9 | 62.3 | 60.7 | 59.2 | 57.8 | 56.4 | 55.0 | 53.7 | 52.4 | 51.1 | 49.9 | 48.8 | 47.7 | 46.6 | 45.6 | 44.5 | 43.6 | 42.6 | 41.7 | 40.9 | 40.0 | 39.2 | 38.4 | 37.7 | 36.9 | 36.2 | 35.5 |
| 85 | 88.2 | 86.4 | 84.7 | 82.9 | 81.2 | 79.5 | 77.7 | 76.1 | 74.4 | 72.8 | 71.2 | 69.6 | 68.0 | 66.5 | 65.0 | 63.6 | 62.2 | 60.8 | 59.5 | 58.2 | 56.9 | 55.7 | 54.5 | 53.3 | 52.2 | 51.1 | 50.0 | 49.0 | 48.0 | 47.1 | 46.1 | 45.2 | 44.3 | 43.5 | 42.6 | 41.8 | 41.1 | 40.3 | 39.6 | 38.9 |
| 90 | 88.4 | 86.8 | 85.3 | 83.7 | 82.1 | 80.6 | 79.0 | 77.5 | 76.0 | 74.5 | 73.1 | 71.6 | 70.2 | 68.8 | 67.5 | 66.1 | 64.8 | 63.5 | 62.3 | 61.0 | 59.8 | 58.7 | 57.5 | 56.4 | 55.3 | 54.3 | 53.2 | 52.2 | 51.2 | 50.3 | 49.4 | 48.5 | 47.6 | 46.7 | 45.9 | 45.1 | 44.3 | 43.6 | 42.8 | 42.1 |
| 95 | 88.6 | 87.2 | 85.7 | 84.3 | 82.9 | 81.5 | 80.1 | 78.8 | 77.4 | 76.0 | 74.7 | 73.4 | 72.1 | 70.8 | 69.6 | 68.3 | 67.1 | 65.9 | 64.7 | 63.6 | 62.5 | 61.3 | 60.3 | 59.2 | 58.2 | 57.1 | 56.2 | 55.2 | 54.2 | 53.3 | 52.4 | 51.5 | 50.7 | 49.8 | 49.0 | 48.2 | 47.4 | 46.7 | 45.9 | 45.2 |
| 100 | 88.7 | 87.4 | 86.2 | 84.9 | 83.6 | 82.3 | 81.1 | 79.8 | 78.6 | 77.4 | 76.1 | 74.9 | 73.8 | 72.6 | 71.4 | 70.3 | 69.1 | 68.0 | 66.9 | 65.8 | 64.8 | 63.7 | 62.7 | 61.7 | 60.7 | 59.8 | 58.8 | 57.9 | 57.0 | 56.1 | 55.2 | 54.3 | 53.5 | 52.7 | 51.9 | 51.1 | 50.3 | 49.6 | 48.8 | 48.1 |
| 105 | 88.8 | 87.7 | 86.5 | 85.3 | 84.2 | 83.0 | 81.9 | 80.8 | 79.6 | 78.5 | 77.4 | 76.3 | 75.2 | 74.1 | 73.0 | 72.0 | 70.9 | 69.9 | 68.9 | 67.9 | 66.9 | 65.9 | 64.9 | 64.0 | 63.1 | 62.1 | 61.2 | 60.3 | 59.5 | 58.6 | 57.8 | 56.9 | 56.1 | 55.3 | 54.6 | 53.8 | 53.0 | 52.3 | 51.6 | 50.9 |
| 110 | 88.9 | 87.9 | 86.8 | 85.8 | 84.7 | 83.7 | 82.6 | 81.6 | 80.5 | 79.5 | 78.5 | 77.5 | 76.5 | 75.5 | 74.5 | 73.5 | 72.5 | 71.6 | 70.6 | 69.7 | 68.7 | 67.8 | 66.9 | 66.0 | 65.1 | 64.3 | 63.4 | 62.6 | 61.8 | 60.9 | 60.1 | 59.3 | 58.6 | 57.8 | 57.0 | 56.3 | 55.6 | 54.9 | 54.1 | 53.5 |
| 115 | 89.0 | 88.1 | 87.1 | 86.1 | 85.2 | 84.2 | 83.2 | 82.3 | 81.3 | 80.4 | 79.4 | 78.5 | 77.6 | 76.6 | 75.7 | 74.8 | 73.9 | 73.0 | 72.1 | 71.3 | 70.4 | 69.5 | 68.7 | 67.9 | 67.0 | 66.2 | 65.4 | 64.6 | 63.8 | 63.0 | 62.3 | 61.5 | 60.8 | 60.0 | 59.3 | 58.6 | 57.9 | 57.2 | 56.5 | 55.9 |
| 120 | 89.1 | 88.2 | 87.3 | 86.4 | 85.5 | 84.7 | 83.8 | 82.9 | 82.0 | 81.2 | 80.3 | 79.4 | 78.6 | 77.7 | 76.9 | 76.0 | 75.2 | 74.3 | 73.5 | 72.7 | 71.9 | 71.1 | 70.3 | 69.5 | 68.7 | 68.0 | 67.2 | 66.4 | 65.7 | 65.0 | 64.2 | 63.5 | 62.8 | 62.1 | 61.4 | 60.7 | 60.1 | 59.4 | 58.7 | 58.1 |
Chart: Lean Angle, Speed, and Turn Radius
If you would like to see a complicated version of the relationship, see Lean Angle, Speed, and Turn Radius: A Complicated Relationship.
If you would like to see the simplest version of the relationship, see Lean Angle, Speed, and Turn Radius: The Simplest Version of the Relationship.