1. Fighter Pilot Math

2. Plan for the Presentation
If you would like, please offer one of these problems to your students as homework or for class discussion.
I’ll talk about each problem briefly during the session and try to interact with whichever classes did that problem.
We won’t get too bogged down in the math during the session so that we have plenty of time for Q and A.
Aside (if you noticed I made an error please let me know so I can correct)

3. Problem 1: Fuel Burn
A F-16 is returning to base with 2000 pounds of fuel on board.
The pilot sets a maximum range fuel flow of 3000 pounds per hour (pph) and the aircraft begins to cruise a .85 Mach
The base is 100 NM away, will the pilot make it with a 100knot headwind?
(Assume a 600knot speed of sound)

4. Discussion
This is a problem that has to be solved inflight quickly and without pencil and paper.
We use some shortcut and approximations – the biggest being the use of Mach
The speed of sound is variable with temperature but we frequently say its 600knots (600 NM per hour). This means that Mach 1 = 600knots or 10NM/minute.

5. How to solve that while flying
1.0 Mach is about 10NM/Min
60 knots of headwind is approx. .1 Mach
.85 Mach (Headwind) = .7 Mach or 7NM/min
100 NM at 7NM/Min is about 15minutes.
¼ of 3000pph is about 700 pounds.
1300pounds at the field. Perfect!
So much for “but I’ll always have a calculator (or my phone)”

6. Problem 2 Crosswinds
After fun mission Ranger 01 is 10 miles from base. The automated weather reports that runway 01 is active, the winds are 055 at 30 knots, and the runway is wet.
The F-16 has a maximum wet runway crosswind limitation of 25 knots. Can Ranger 01 land at base?

7. Discussion
This is another one of those mental math problem that we face regularly. Again we have to resort to approximations to solve this inflight.
Despite the approximations a pilot really needs to understand the underlying trig.
45 degrees

8. How to solve that while flying
Runway 01 is oriented on a 010 heading, the wind is 45 degrees off.
A approximation for Sin 45 is .7
.5 of 30 is 15.
.2 of 30 is 6.
21 knots ok we’re good. (Good thing or we’d have another fuel calculation for our divert)

9. Problem 3 Low Level:
Ranger 01 is flying a F-16 in Alaska below the mountains at 500knots. – which is awesome by the way!
He needs to make a 90 degree turn in a 1/2 mile wide canyon
Ranger 01’s Fuel Tanks are limited to 7.0 G’s.
Can Ranger 01 make the turn without exceeding the G limits?
If not – what could Ranger 01 do?

10. Discussion
If flight this problem is a mix of the academic and the athletic. The pilot needs to understand the underlying math while also being able to recognize the problem by looking out the window.
G forces are the (reaction to) radial acceleration: a=v^2/r
G’s are multiples of gravity’s 9.8m/s^2

11. More Discussion 500Knots is 257 m/s 1/2 (statue) mile is 804m
R= (257m/s)^2/(7x9.8m/s^2) = 962m
Nope!
So what’s a pilot to do?
Slow down!
350knots is 180m/s. At that speed the radius is 470 m.

12. How this works in practice
So we’re not out doing this math while flying
But each pilot understands the relationship between speed, G’s and turn radius.
Most pilots have some rules of thumbs when they mission plan. But in flight a pilot has to see it, understand what is going on and react appropriately

13. Problem 4: Ballistics Ranger 01 needs to drop a bomb into a cave.
The optimum angle for the bomb is 36 degrees
The weather is bad and Ranger can only be 3000’ above the cave
How fast does Ranger 01 need to fly?
Ignore air drag on the bomb

14. The bomb triangle D=1/2gt^2 2D/g = t^2 T=13.6s Vy=gt=133m/s
Tan 36 = 133/Vx
Vx = 183m/s = 355knots
36 degrees
Vy the velocity due to gravity
Vx – Velocity the Jet imparts to the bomb

15. Discussion
We use computers to solve this but the pilot needs to understand the why and how. Because garbage into the computer….
The impact angle of the bomb is function of the aircrafts speed and the altitude of release
At a constant altitude, impact angle is inversely proportional to the jet’s airspeed.
But, If the weather was lower than forecast the aircraft would need to slow down
36 degrees
Vy the velocity due to gravity
Vx – Velocity the Jet imparts to the bomb

16. Problem 5 Fuel Burn
I have 1,000 pounds of fuel remaining. If I’m burning 2,500 pounds per hour how much longer can I fly?

17. Discussion 2,500 lb/hr x T = 1,000lb T=1,000lb /2,500lb/hr
T=.4hr, approximately 25 minutes.

18. Problem 6 Landing Weight
At takeoff the jets weighs 36,000lbs; 12,000lbs of which is fuel. I want to land no heavier than 28,000lbs. How long must I fly if I’m burning fuel at 4000 lb/hr ?

19. Discussion W1-(RT)<W2 36,000lb - (4000lb/hr T) < 28,000
T = 2hr