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How quickly can a flying character turn?

A flying character can turn by changing the direction of their thrust. Turns can be described by a quantity called the turning radius. Imagine that a character continued his turn until he traced out a full circle by coming back to his starting point; the radius of that circle would be the turning radius for that particular turn. The character doesn't have to go in full circles; a 90 degree turn is a quarter circle, a 180 degree turn is a half circle, etc.... Circular turns are the most efficient way for a character to change their direction.

In order for an object to be turned there must be a force exerted on it towards the center of the circle described above. As the exerted force gets stronger characters can make tighter (smaller radius) turns. For the following calculations I will typically assume that the character is exerting all of the thrust he has available to make the turning radius as small as possible.

However this force is exerted, a character will turn at constant velocity when the total inward force (also called the centripetal force) has a magnitude of:

Ftotal = m v2 / rturn
(total, centripetal force) = (character's mass) * (character's velocity)2 / (turning radius)

Characters will generally have three forces acting on them while they are flying: gravity pulling them downwards, air friction slowing them down, and their own thrust. If we add these forces together so that the net force gives us a centripetal force in the direction we wish to turn (and perpendicular to the direction we're travelling) we will get the most efficient use of our thrust. In the case of a purely horizontal (left or right) turn:

turn_side.JPG (14436 bytes)

The character's thrust has to be exerted upwards to cancel the downward pull of gravity, as well as forwards to cancel the horizontal force due to air friction and sideways to provide the total force required to make a turn. This means that:

Fthrust2 = Ftotal2 + Fgrav2 + Ffriction2
Fthrust2 = ( m v2 / rturn )2 + ( m g )2 + ( cfriction v2 )2

rturn = m v2 / [ Fthrust2 - ( cfriction v2 )2 - ( m g )2 ]

This equation seems complicated, but with a little bit of algebraic work it will become much more simplified. Remember that we determined the values of Fthrust using the relation:

Fthrust2 = ( cfriction vmax2 )2 + (m g)2

We are always be able to express a character's velocity as a fraction of his maximum velocity, as follows:

v = N vmax

The value of N will always be between zero and one. Ex: If a character is traveling at 95% of his maximum velocity we will use N = 0.95.

When we substitute these expressions for Fthrust and v into the expression for rturn we obtain the following:

rturn = m v2 / [ Fthrust2 - ( cfriction v2 )2 - ( m g )2 ]
= m v2 / [ ( cfriction vmax2 )2 + (m g)2 - ( cfriction v2 )2 - ( m g )2 ]
= m v2 / [ ( cfriction vmax2 )2 - ( cfriction v2 )2 ]
= m v2 / [ cfriction2 ( vmax4 - v4 ) ]
= m N2 vmax2 / [ cfriction ( vmax4 - N4 vmax4 ) ]
= m N2 vmax2 / [ cfriction ( 1 - N4 ) vmax2 ]

rturn = m N2 / [ cfriction ( 1 - N4 ) ]

The value of rturn is independent of a character's maximum velocity and maximum thrust (and is therefore independent of what super ability gives them flight, their experience level, etc...)! The only variables that will change between different characters are their mass and their frictional variables. Assuming that most human sized objects have the same cross-sectional area and general shape we can calculate a character's minimum turning radius knowing only his mass and velocity! One final simplification we can make is to lump the N terms together into one part, called eN. The value of eN will be independent of mass, power and experience levels, as well as maximum velocity.

rturn = m eN / cfriction

eN = N2 / ( 1 - N4 )


Results

We will make the same assumptions about the frictional force variables as when we determined the maximum available thrust: cfriction = ( 0.08125 kg/m). To get some sample values for rturn I will assume that my character weighs 250 lbs ( m = 113.4 kg ), and is carrying no additional mass. For this character, exerting his maximum thrust in the ideal direction (as shown in the diagram above), the minimum possible turning radii for several different fractions of his maximum velocity are:

N 
99%
90%
75%
50%
25%
10%
5%
1%

eN
4.9374
1.3812
0.68033
0.25820
0.06262
0.01000
0.00250
0.00010

  rturn
6891 m
1927 m
949.5 m
360.4 m
87.4 m
14.0 m
3.49 m
0.14 m

Another important consideration to take into account is the time required to make these turns. This time can be calculated as:

tturn = 2 pi rturn * ( a / 360 ) / v
tturn = 2 pi ( a / 360 ) m eN / ( cfriction v )

where a is the angle swept out by the turn. A right or left turn is 90, turning around is 180, a full circle is 360, etc. At 90% speed it will take a character with Sonic Flight 10.8 seconds to make a 90 turn, while it will take a first level character with Flight: Wingless 34.2 seconds for the same turn. While travelling at 10% speed it will take the Sonic character 0.70 seconds and the first level Wingless character 2.23 seconds to make a 90 turn. These calculations tell us that travelling at high speed isn't always the quickest way to get somewhere; slowing down to make a turn is just as wise when flying as it is when driving a car, riding a bike, piloting a hover craft....