You're flying solo in a late-model, fixed-gear Cessna 172 with half-full fuel tanks. You are at 1000 feet AGL, 100 knots indicated, directly above the middle of a 2000 foot long straight section of paved road. There is a steady 15 knot wind blowing up the 10 degree, wide, flat and clear slope that the road runs straight up. Your engine suddenly seizes, prop horizontal, with no possibility of restarting it. Everything else about the aircraft is perfectly functional and the height of the terrain is not significantly above sea level for performance purposes. The road section heading is 360 degrees, due north up the hill. Choose any initial heading for the engine failure and remember it occurs directly over the mid-point of the road section (runway). Please provide information on the specific flight segments in terms of speed, turn rate, descent rate, etc. and consider the effects of the wind blowing directly upslope (without any crosswind component with respect to the roadway). Explain why your answer is (nearly) optimal. How do you make the safe landing?
Admittedly, the 10 degree sloped roadway is an anomaly, and you would never see an actual runway built on this grade. Don’t try to apply any of this logic to a shallower runway without some careful analysis.
TLDR: I favor the uphill/downwind landing because it gives you more time for maneuvering and requires less pilot skill to manage the roll-out after landing.
Here's my take on the possible landing approaches. I think my performance estimates are generally conservative. I think the flight maneuvers are fairly pedestrian until landing and roll-out. Your mileage may vary. These are not optimal, but I think it is possible to land either uphill or downhill with varying degrees of risk and maneuver aggressiveness from nearly any initial heading. Obviously, some headings significantly reduce the risk of one direction over the other. In half of all cases you may have to make right turns instead of the depicted left turns.
Here are my assumptions:
Altitudes are all referenced to the roadway midpoint elevation
Aircraft weight is 2000 lbs.
40 KIAS -- stall in landing configuration
45 KIAS -- 1320 ft/min descent - 30 degrees flaps
48 KIAS -- stall in clean configuration
50 KIAS -- 780 ft/min descent - 10 to 20 degrees flaps
55 KIAS -- 725 ft/min descent
60 KIAS -- 800 ft/min descent
65 KIAS -- 870 ft/min descent - best glide angle through the air approx. 7.5 to 1 glideslope
70 KIAS -- 970 ft/min descent
75 KIAS - 1115 ft/min descent
80 KIAS - 1285 ft/min descent
The wind is blowing from the South at 15 knots
Horizontal wind component up the slope: ~1500 ft/min
Vertical wind component (upward) up the slope: ~264 ft/min
Using your cat-like reflexes and finely-honed piloting skills, track outbound on a heading of 235 for 16 seconds while trading your 100kts of airspeed for a little more altitude. I calculate you can pop up about another 80 feet from the mid-field starting altitude during that time.
Fly for an indicated airspeed of 70 kts because you must now turn (~1.1G load factor) at 6 degrees per second to a heading of 180. This takes about 9 seconds; you are 1600 feet south, 2280 ft west, and about 950 ft above mid-field. 26 seconds have elapsed.
Fly on a heading of 180 for nearly 40 seconds. Think about making a radio call stating position and situation. Maintain 70 KIAS for better penetration into the wind. You are in your downwind leg and it has been over 1 minute since your engine failed. You are 5230 ft south, 2280 ft west, and 495 ft above mid-field. About 65 seconds have elapsed.
Now turn, maintaining 70 KIAS, at 6 degrees per second to the "runway" heading of 360. Take a few extra seconds to roll out and line up with the roadway. You are about 3935 ft south (thanks to that upslope wind), aligned with the centerline of the road, and 108 ft above midfield (about 750 ft AGL because of the slope). About 97 seconds have elapsed.
Slow to 65 KIAS and fly this heading for about 27 seconds. Wait until the end to drop maybe 10 degrees flaps to preserve speed for the zoom-climb, flare landing. Nose-high attitude will be your friend on the upslope landing so don't use too much flap deflection. About 114 seconds have elapsed.
You cross over the landable portion of the road at 65 KIAS with over 100 ft AGL. It has been about 118 seconds since your engine failed. Fly the aircraft through an energy-preserving flare (actually climbing at the last) until a few feet above the rapidly rising road. You will touch down just past the first third of the landable roadway a little over 2 minutes since your engine failed. Your speed will bleed off, the airplane will stop flying, and because of the upslope, you will roll to a stop, with or without brakes in 10-20 seconds. Apply brakes so you don't roll backward down the hill!
Track outbound from the midfield point on a heading of 90 degrees for 7 seconds while trading airspeed for altitude. The maneuver takes about 15 seconds to complete, but half-way through, begin a 6-degree-per-second turn, stabilizing at 70 KIAS and rolling out on a heading of 360.
Fly the 360 heading for about 4 seconds. You are now about 2340 feet north, 2255 ft east, and about 925 ft above the midfield point. 26 seconds have elapsed.
Maintaining 70 KIAS, begin a 30-second, 6 degrees-per-second turn to heading 180. You are now about 2940 ft north (upslope wind again), aligned with the roadway, and 550 ft above the middle of the roadway. Your height AGL is only about 65 ft (!), but that’s okay. You are on final approach and you can fly into and out of ground effect as you like to fly to the threshold of the landable portion of the roadway which is on the steeply descending slope. It has been about 56 seconds since your engine failed.
Just under 60 seconds have elapsed. At this point you need to increase your rate of descent to more than the 10 degrees (17% gradient) downslope of the hill and roadway. You might choose to drop 30 degrees flaps and decrease airspeed to 45 KIAS and plunk down on the roadway at about 30 knots groundspeed, but you have to bleed off 35 knots of airspeed without ballooning too much and end up at only about 5 knots above stall. Your angle of attack would be so high that you would not be able to see the roadway at all - not even the far end because of the down slope. And you would strike the tail on the ground before the main wheels touched.
Instead, you will need to increase your airspeed to about 80 KIAS and use about 30 degrees of flaps to keep from flying up and away from the ground. This feels like a fairly steep approach but you are only descending at about 250 ft/min with respect to the ground which is falling away steeply. You are about 2000 feet from the landable portion of the runway. In about 20 seconds you are down and on the roadway about 100 yards down from the start of the landable portion. About 1 minute and 17 seconds have elapsed since your engine failed.
Here's where things start to get a little sketchy.
You are rolling down the roadway with 80 knots indicated (65 kts ground speed) – give yourself 6-8 seconds to get the flaps fully retracted while you try to keep the aircraft “stuck” on the ground. On a 10 degree downslope your brakes are (cosine 10 degrees) still about 98% as effective, if you weren’t still generating a lot of lift. Conservatively, you might get 50% of your weight onto the wheels initially. And don’t forget your mass is effectively pulling you down the slope (sine 10 degrees * 2000 lbs) with about 345 pounds of force.
For reference, this source: (https://aviation.stackexchange.com/questions/67979/is-there-an-actual-rated-thrust-for-a-cessna-172-engine-at-takeoff-power#:~:text=As%20John%20K%20said%2C%20it,be%20aware%20of%20before%20takeof) puts the thrust of the 160hp, fixed pitch 172N at between 500-600 lbs force when at takeoff power.
You will trade lift at higher speeds of the roll-out, which will make your brakes less effective, for the drag at the higher speeds which will counter-act the downhill force.
It’s probably a 30+ second, white-knuckle ride down the roadway managing your energy, trying to keep the plane “stuck” on the ground and to not lock up the lightly loaded mainwheel brakes until you get it stopped just before the end of the useable portion.
I conclude that the uphill/downwind landing is the better choice for the following reasons:
- Time. Compared to the downhill/upwind landing you have about 30 seconds more seconds to collect your thoughts, take calming breaths, and possibly make a radio call reporting your position and situation on one of the straight legs of your “pattern”. You buy the extra time by landing on the end of the roadway that is almost 350 feet lower than the top end.
- Distance. Right up until touchdown, the proximity of the ground is not a factor at any part of the uphill/downwind approach. Contrast this with virtually skimming the ground for 15 seconds at the last of the downhill final.
- Energy. Granted, change in energy on the uphill approach happens much more quickly at the landing, but because you are flying up the slope there is some leeway on the timing and aggressiveness of the landing flare. And once you are down, you will roll to a stop even if you don’t use the brakes. On the downhill landing both your kinetic energy and your potential energy are working against you getting your aircraft stopped. The relatively feeble upslope wind does little to help you here, and in fact, hurts you once down by producing more lift and reducing the weight on wheels for braking. There will likely be an excruciating 30 seconds of managing flaps, brakes, and elevator so you can brake to a stop before the end of the useable roadway.
UPDATE: Long edit coming that verifies the downhill/upwind landing is a viable option.
TLDR: @Robert DiGiovanni has a really good point about using forward slips to control speed/energy in steep descents.
It's been more than a month but I finally got a chance to go flying and experiment with forward slips. We had three people in the "steam-gauge" panel Cessna 172S, including a very experienced instructor pilot from one of the local FBO's. 3/4 tanks on both sides had us at about 2450 lbs on the ramp. We didn't get as much data as I had hoped due to an abundance of caution and other pressing purposes for the flight, but here's what we found:
30 degrees flaps, steady-state, idle power glide - about 650-700 ft/min descent
20 degrees flaps, full rudder forward slip, ailerons to maintain heading - about 1000 ft/min descent
30 degrees flaps, full rudder forward slip, ailerons to maintain heading - about 1300 ft/min descent
The rest of these cases are 30 degrees flaps, full rudder forward slip, ailerons to maintain heading
65 KIAS: ~1600 ft/min descent
70 KIAS: ~1800 ft/min descent
75 KIAS: 2000+ ft/min descent
80 KIAS: Not brave (or foolish?) enough to fly this close to the top of the white arc in a full slip where airspeed readings are inaccurate and delayed!
We often could not establish (or accurately sustain) exact airspeeds, and also, as we all know on some level, the response of the airspeed indicator and vertical speed indicator are both inaccurate and significantly lag (~7 seconds for our VSI) the actual aircraft state, especially in a full rudder slip. I tried to use (potential and kinetic) energy analysis methods to compensate for some of the lag but the data set was small enough and noisy enough that I judged it better to estimate the time lag and take appropriately delayed readings from the vertical speed indicator. This said, even though the VSI read 0 on the ground, I couldn't trust it as 100% accurate - so take all this data with a grain of salt.
Other things learned: This airplane doesn't really like to be in a full (30 deg) flaps forward slip. At one point the airspeed indicator read 60 KIAS and the stall horn was intermittently beeping (turbulent air over the port?). In another prolonged slip, with well over half in both tanks, the low fuel indicator blinked a couple of times for the higher trailing wing. I was also surprised to see the vertical speed indicator read only about 500 ft/min descent at 68-70 KIAS throughout a standard rate turn. I verified via my video of the panel that we descended about 800 ft in one minute in the mostly steady-state turn. It should theoretically be more like 900 ft/min in still air. The air seemed smooth, but we may have had weak, broad lift over the large decommissioned airbase below us - and maybe the vertical speed indicator was no longer well-calibrated.
So to sum up, there were lots of potential sources of errors, but the data held no big surprises. It verified @Robert DiGiovanni was right that you can get controlled, high rates of descent by using forward slips. The downhill/upwind landing becomes much more manageable. The 65 KIAS, 30 degrees flaps case he suggests gets you a descent rate into the headwind that gives you a 15.3 degree descent angle. If you can wrap your head around this, use this like a (15.3 - 10) 5.3 degree glideslope or less to get to the threshold. Use less than full rudder slip for a shallower approach angle and the eventual 'flare'. If you can keep the airplane stuck on the ground it was the equivalent of a ~50 knot touchdown onto the road surface (but going downhill though), not the 80 kts that my approach case got you. I would still pick the other direction, but the downhill/upwind option would definitely work, and in the hands of a fairly skilled pilot, leave you an airplane without a scratch on it.
As for that seized engine that got you here in the first place? That's a problem for when you get the airplane back off the mountain road and into a mechanic's shop.
Unless the wind is too strong you should try to land uphill.
Tailwind will increase the required length of the runway.
When you land uphill you have a significantly larger angle between your glide path and the hill slope compared to landing on a horizontal runway. That means the required length of the runway decreases.
Both effects work against each other so that (depending on the strength of the wind) the resulting required length of the runway is approximately the same as landing on a horizontal runway with no wind.
When you land downhill you have a significantly smaller angle between your glide path and the hill slope compared to landing on a horizontal runway. That means the required length of the runway increases significantly.
When additionally landing against wind downhill the oncoming wind reduces the ground speed and thus increases the angle between your glide path and the hill slope.
But the wind coming uphill also generates additional lift (known as ridge lift) that flattens your glide path even more and thus the angle between the hill slope and your glide path is decreased even more so that it is questionable that you even touchdown before you reach any obstacles at the end of the hill.
For weak and medium wind speeds the combined effect of down slope and ridge lift is stronger that the tailwind effect at uphill landings (depending on the slope angle and wind speed).
Disclaimer This only applies if you cannot touchdown at the hill top and your touchdown point would be on the downhill slope.
Tailwind landings uphill are common on airfields with a sloped runway. One famous example is the Wasserkuppe air field in Germany:
This Youtube video shows a pattern flight at the Wasserkuppe airfield starting downhill against the wind and landing uphill with tailwind. When the pilot starts the engine on the right you see a glider landing downhill against the wind.
Landing downhill/into the wind may result in more wear on the brakes if done often, but would be much safer as an emergency procedure, especially in unfamiliar terrain.
A 10 degree downslope on a road would be a 17% down grade, a very rare road indeed. The glide slope of a 172 is around 1:8 which works out to a 7 degree downslope (using sin and arcsin trigonometry), which is a 12.5 % downgrade, what to do?
With an emergency landing, it is best to land into the wind. A crash at 30 knots groundspeed is far more survivable then at 70 knots. Because of weight considerations, the 172 is far more fragile in a collision than a modern car. Walking away from a 70 knot crash is highly unlikely.
Keep in mind, in a steady state glide, the component of gravity pulling you forward, sine(glide angle) x mass, is equal to your drag and should not dissuade one from landing at a steep angle.
Gravity will only increase the roll-out distance as the plane slows down after touchdown, but this is offset by much lower ground speed when landing.
For the downwind/uphill landing drag and gravity now work together to rapidly slow the aircraft, but care must be taken as the groundspeed, therefor time over runway, is much less. In this particular case, calculating 1/2mv$^2$ vs mgh at 65 knots (33 m/s) will work well:
Kinetic Energy from Speed = 900 kg/2 × (33m/s)$^2$ IAS = 490,000 kg (m/s)$^2$
Potential Energy from Climb = 900 kg × 10m/s$^2$ × 100m = 900,000 kg (m/s)$^2$
even without factoring in the drag contribution:
Drag over Runway = 900kg/2 × 33(m/s)$^2$ approach speed - 900kg/2 × 23(m/s)$^2$ landing speed = 252,000 kg(m/s)$^2$
or ground effect. As with all landings, too high an approach speed will carry the risk of overshooting the runway, with 15 knots tailwind added.
1000 feet AGL is plenty to do a 180 degree turn with 300 - 500 feet to spare if you follow your procedures and get to best glide speed first. Remember, into a headwind, this IAS will be slightly faster than Vbg.
Now for that glideslope. The solution is exactly the same as with landing a high performance glider: increase your angle of descent by reducing your lift/drag ratio. You would go to flaps 30 and/or full forward slip.
A full forward slip in a 172, into the wind, greatly increases the glideslope. To land, round out, lower AOA by reducing pitch, and hope your brakes work (They should at least once at 30 knots groundspeed even with the grade). Amazingly, this is one case where the older Cessna 170 taildragger would actually do a bit better by doing a "wheels" landing (without risking a nose wheel strike).
Another option would be to use the other end of the drag bucket by slowing down and increasing AOA, but under windy conditions this is much more risky.
Thankfully, even the steepest road grades (4-8%) work out to less than 5 degrees of slope.
Although the expert pilot may prefer to "stick" the landing uphill, the chances of completely missing a downwind approach make it unadvisable with that much tailwind.
If there was no wind, I'd probably land uphill, but in the circumstance you describe I'll take the into-wind direction landing down the hill. This gives the lowest ground speed and therefore the lowest energy state if things don't go so well (you have a 30 knot energy difference between the two landing directions - that's a lot, and a bigger deal all around than the uphill/downhill aspect).
The into-wind glide slope being steeper than the down wind glide slope, the headwind negates some of the problem of having to achieve a steeper decent angle to touch down on a down sloping hill, and you can make a 172 come down pretty steeply anyway if you really have to. If I misjudge the whole thing trying to land into wind down the hill, and don't touch down until the far end, I roll into the bushes doing 40 kts at ground contact, instead of 70 kts if I overshoot going the other way.