The non-afterburning turbojet engines of a Cessna T-37 jet trainer produce approximately 1,700 lb of installed thrust for takeoff at sea level. Its takeoff weight is 6,575 lb. Its for takeoff, and its drag
polar in its takeoff configuration is . Its reference planform area is 184 ft2. Normal
takeoff procedure requires the pilot to rotate the aircraft to the takeoff attitude just prior to reaching takeoff velocity. Therefore, for the majority of the takeoff roll, the aircraft’s . What will be the aircraft’s takeoff distance at sea level with no wind?
Example 5.10
Page 5 of 11 Unit C-3
Landing Distance for Conventional Takeoff & Landing (CTOL)
• The landing maneuver is broken up into approximately the same steps as takeoff (in reverse).
• As with takeoff, the details of the design requirements for landing distance vary. The landing speed
( )LV is usually specified as stall1.3V .
• The approach (or decent) to landing is also at LV or slightly faster.
• The forces of landing are similar to those for the takeoff, except that thrust is zero and (now
called "braking coefficient") has much higher value (because brakes are applied). Braking
coefficient values are 0.4 – 0.6 (for dry concrete), 0.2 – 0.3 (for wet concrete), and 0.05 – 0.1 (for an
icy runway).
• The same analysis steps as for takeoff yield:
( )max
2
0.7
1.69
L
LL
L L V
Ws
SC g D W L =
+ − (5.59) landing distance
• Note that "1.69" for landing, instead of "1.44" for takeoff, is because stall1.3LV V= , whereas
TO stall1.2V V= . As with the average force values for takeoff, the average deceleration forces are
evaluated at 0.7 LV .
Takeoff Abort Distance
• A common design requirement is for an aircraft to be able to safely use a runway of a specified
length. For a single-engine aircraft, this means that the plane must be able to accelerate to takeoff
speed, have the engine fail at that point, and then abort on the remaining runway.
• The calculation for the total distance required for this maneuver includes a normal takeoff distance
calculation, some allowance for reaction time (usually 3 seconds), and a calculation for braking
distance.
• Because braking starts at takeoff speed (rather than landing speed), the factor multiplying the
breaking distance calculation in eq(5.59) is 1.44 instead 1.69.
Landing
(5.59)
Page 6 of 11 Unit C-3
Turns
• The most important characteristics of turning performance that are frequently specified as design
requirements are turn rate and turn radius.
• A sustainable turn rate (or turn radius) is performance that aircraft can maintain for a long period of
time (minutes, or even hours).
• An instantaneous turn rate (or turn radius) is a capability that aircraft can achieve momentarily, but
then the maximum performance might begin to decrease immediately.
Level Turns
• In level turn maneuver, the aircraft maintains a constant altitude (and in a sustainable turn, a constant
airspeed). Its velocity vector changes directions but stays in a horizontal plane.
• Summing forces in the vertical direction yields:
vert 0 cosF L W= = − => cosW L = (5.60)
: Bank angle of a level turn
=> 1 cos L W n = (5.61) the load factor
• The load factor "n" is alternatively known as the "g" force that the aircraft is pulling, but the symbol
g is already being used to denote the acceleration of gravity.
• Note that the acceleration in the vertical direction is set to zero, because the aircraft's motion is
assumed to remain in a horizontal plane.
Level Turns
(5.60)
(5.61)
(5.62)
(5.63)
(5.64)
Page 7 of 11 Unit C-3
• Summing forces perpendicular to the velocity vector in the horizontal plane gives: 2
2 2 2 2
horiz sin 1W V
F mV r L L W W ng r
= = = = − = −
=> 2
2 1V
ngr
= − which can be solved for the turn radius r: 2
2 1
Vr
g n=
− (5.62)
• The result is independent of aircraft type (Boeing 747 and F-16 at the same bank angle and airspeed
will have the same turn radius). The rate of turn is V r = , so:
2
2 2
1
1
V g n
VV g n
−= =
− (5.63)
Drag for a Turning Aircraft
• Summing forces parallel to the aircraft's velocity vector yields:
W dVF ma T D
g dt= = = − which sustained turn simplifies to: T D−
• This seems to be a simple result, but drag for a turning aircraft is not the same as for one in SLUF.
Most of the additional drag in a turn results from additional induced drag caused by the additional
lift required to turn.
• Recall eq(5.61) that L nW= , and so:
L
nWC
qS= =>
( )0 0
2 2
D D
k nWnWD C qS qSk C qS
qS qS
= + = +
(5.64)
• Induced drag in a turn increases with 2n . This can make it very difficult for many aircraft to sustain
Two different airplanes: one on Mars and one on Earth, are performing level turns at identical true airspeeds of 100 m/s and identical bank angles of 60 degrees. How do their load factors, turn radii, and rates of turn compare?
Example 5.11
Page 9 of 11 Unit C-3
Vertical Turns
• A simplest vertical turn model is a turn made purely in a vertical plane (i.e., 360 turn from the
original flight path: a loop).
o The maneuver will start with a pull-up (a vertical turn from initial straight-level condition).
o At the top of the loop, the aircraft is performing a pull-down from the inverted flight
condition.
o A pull-down can also be entered from straight-and-level flight by rolling the aircraft
(changing the bank angle ) until it is inverted.
o A vertical turn initiated by rolling inverted and then pulling down, completing the turn to
level flight headed in the opposite direction is known as "split-S."
• For the pull-up, summing the forces perpendicular to the velocity vector yields: 2
2 WVF mV r L W
gr⊥ = = = − =>
2
1 1V
L W ngr
= − = −
( )
2
1
Vr
g n=
− (5.65) and
( )1g n
V
−= (5.66)
• For the pull-down, similarly: 2
2 WVF mV r L W
gr= = = + =>
2
1 1V
L W ngr
= + = +
( )
2
1
Vr
g n=
+ (5.67) and
( )1g n
V
+= (5.68)
• Finally, for the case where the aircraft's velocity vector is vertical: 2
The procedure for performing a loop in the T-37 begins with diving the aircraft to reach the entry airspeed (250 KIAS at 15,000 ft), and then pulling straight up (check wing level) with n = 4. As the nose comes up, the speed begins to bleed off. At about 200 KIAS or a little less you start to feel a tickle in the stick. This tickle is the first signs of stall, and it occurs before the drag due to lift (induced drag: due to wing-tip vortices and flow separation) gets too large. If you pull into heavy buffet at this point, the excessive drag will cause the aircraft to slow down too much, and you will not make it over the top. So, you pull on the tickle, relaxing back-pressure on the stick as the aircraft slows down with n = 3: this is at “side of the loop” (180 KIAS at 18,000 ft). As the pitch attitude exceeds 90 degrees, you throw your head back and look to find the horizon, and then pull down to it, being careful to keep the wings level. If you do everything correctly, you will make it to the top of the loop with n = 1.5 (100 KIAS at 19,000 ft). Then, as you continue to pull down inverted, the speed builds up, as do the back-pressure and load factor required to stay on the tickle. When the load factor reaches 4, you hold that until you complete the maneuver. From the cockpit, a T-37 loop flown as just described looks like a circle, as shown in Fig. 5.39. But what would it look like from the ground or from another aircraft? Calculate the turn radius at the three points shown, and then to estimate the actual shape of the T-37’s flight path as it performs a loop.
