A mountain departure that cannot meet the ODP gradient, plus an out-of-date transponder
You are departing a mountain airport (field elevation 5,900 ft MSL) in a normally aspirated PA-28-181, maximum gross weight 2,550 lb, on an IFR clearance to the east. Runway 17's published obstacle departure procedure requires a minimum climb gradient of 320 ft/NM to 10,500 ft before proceeding on course. The surface temperature is 28 C with the altimeter 30.12, so the pressure altitude is essentially field elevation. With two adults and 48 gal of fuel you weigh 2,400 lb. Your performance chart shows 650 fpm at 80 KIAS at sea level ISA and about 330 fpm at 80 KIAS at a density altitude of 9,000 ft. Runway 35, the opposite direction, publishes a 250 ft/NM gradient and has a 15-kt tailwind component today. Your clearance will take you to 12,000 ft. During the preflight you notice the transponder was last inspected 26 months ago; the avionics shop says it 'works fine' but cannot fit you in until next week.
1. Compute the density altitude and your actual climb gradient. Can you legally accept the Runway 17 departure?
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ISA temperature at 5,900 ft is about 3 C, so 28 C is roughly ISA +25. Density altitude is approximately pressure altitude plus 120 times the ISA deviation: 5,900 + 120 x 25, about 8,900 ft, call it 9,000. At that density altitude the chart gives 330 fpm at 80 KIAS. In still air, 80 KIAS is 1.33 NM per minute, so the gradient is 330 divided by 1.33, about 248 ft/NM. The procedure requires 320 ft/NM, which would take roughly 430 fpm at 80 kt. You are about 22 percent short. The gradient is a legal minimum, not a target, so you cannot accept a clearance that routes you over the Runway 17 ODP in these conditions. Options are to wait for a cooler departure, reduce weight, use a different runway, or depart VFR if the weather and terrain allow. Be careful with the cooler-morning option: at 15 C the density altitude drops to about 7,300 ft, but you must re-run the chart rather than assume the gradient is now satisfied, and on this airplane it may still fall short.
2. The transponder was last inspected 26 months ago. What does that mean for this flight, and what are your options?
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The transponder must be tested and inspected within the preceding 24 calendar months, so at 26 months you may not operate it at all; a shop's opinion that it works fine is not a substitute for the inspection. The next question is whether the flight needs a transponder. Your clearance takes you to 12,000 ft, and a transponder is required above 10,000 ft MSL in the contiguous United States, except at or below 2,500 ft AGL. As planned, the flight therefore cannot be made legally with the transponder inoperative. Options: get the inspection before departure, which the shop cannot do today; restructure the flight to stay at or below 10,000 ft for the entire route and coordinate a transponder-out operation with ATC; or do not go. Note that restructuring is not available here, because the Runway 17 ODP itself climbs to 10,500 ft. If you do fly with the transponder off, tell ATC and expect routing and altitude restrictions, and remember that the flight as originally cleared is not one you can accept.
3. Runway 35 publishes 250 ft/NM and has a 15-kt tailwind. Evaluate that option.
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Climb gradient is feet of climb per nautical mile over the ground, so wind matters. At 80 KIAS with a 15-kt tailwind your ground speed is about 95 kt, or 1.58 NM per minute, and 330 fpm divided by 1.58 gives roughly 209 ft/NM, which is well below the 250 ft/NM requirement. The lower published gradient on Runway 35 does not rescue you; the tailwind makes the actual gradient worse, not better. A tailwind also lengthens the takeoff roll and reduces the obstacle clearance margin in the initial segment. Flip the wind around to see the effect: a 15-kt headwind on Runway 17 would give a ground speed of about 65 kt, or 1.08 NM per minute, for a gradient of about 305 ft/NM. That is closer to the 320 ft/NM requirement but still short. The lesson is to compute the gradient using ground speed derived from the actual wind, not still-air knots, and to compare that number to the published requirement before accepting the clearance.