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Re: How is a pilot supposed to find out their distance in...
Using the old (pre GPS) method you would use the sectional and plotter (aviation ruler) and measure the distance. Using a GPS you can usually turn on distance rings on the screen or if you have the airport selected as destination you can read it right off the screen. If it's your home base you should learn the landmarks and their distances from the airport. In the Miami area, there are so many airports and landmarks, if you're flying there, during your preflight preps you should measure out distances to some landmarks that you plan to pass over and mark them on the chart or flight log.
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It's quite permissible to do so, however a general aviation pilot in a low performance plane should be ready for fast instructions and quite a bit of maneuvering to stay out of the way. The best time to do it would be at night. Here's a youtube video of one doing it. https://www.youtube.com/watch?v=KKvWn317tpU
160 is the final approach heading to runway 18. You notice that this approach does not align exactly with the runway, you make a slight right turn to land. 115deg is the outbound leg heading for a procedure turn and 295 is the return heading from the procedure turn, from which you would turn 160 on final and fly the 160 line to the missed approach point. DME is required for this approach so you would either need DME or a GPS that can supply distance information. To fly this full approach (from southerly directions) you would fly to the VOR, fly outbound at 3000' MSL on a 340 heading for about 7 miles then turn left to heading 115 for 1 minute then right turn to 295 and intercept the 340 radial, turn right to 160 and descend to 2300' by GRAMA and continue descending to 1220" at 1.6 DME from the VOR. From that point, if you have the proper visual cues you may descend below 1220' to land, if you can't see the runway environment from 1220 feet at 1.6 DME then you must stay at that altitude until you do see it or you reach the missed approach point at .3DME. There's a very real reason that an instrument rating is required for IFR flight. It takes a lot of training and practice to learn to do it right - and even then you may not have it all correct. (I may well have missed something on this approach but I haven't been IFR current in several years)
It is very common for small airports not to have a control tower, radar, or communications equipment. There are very well developed procedures for "uncontrolled airports". There is a standard traffic pattern that aircraft fly at almost all airports (consisting of a downwind, base, final, and upwind leg) and there are specific radio calls that are supposed to be made at certain points in the pattern. Most airports have a fixed base operator to supply fuel and services. They often monitor the common traffic frequency and supply some info to pilots about wind direction and runway in use. There's also a specific way to enter the pattern - usually at a 45 degree of the downwind leg, Yes it's possible for aircraft to collide and it happens several times a year - usually when a low wing airplane is above a high wing airplane in the pattern. Neither can see the other so occasionally that can happen.
Technically is you are flying domestic you are not flying over foreign airspace. If flying IFR you must file one. Needless to say when filing a flight plan you must decalre if you are domestic of international flight.
The AIM is not regulatory so there is not really any absolute rules to go by. The AIM does give procedures which should be followed but some pilots do not. In many cases they make it harder for the rest of us who do follow the AIM procedures. I was watching a YouTube video yesterday where a group of "supposedly" accomplished pilots flew a C172 from DAB to CDK. These hotshots proceeded to fly a 10 mile straight in to Cedar Key, not even apparently looking for NORDO traffic, thus ruining the trip as far as I was concerned.
You're stuck with a water landing (ditching). Hopefully at altitude the pilot can see a ship and make the landing in the vicinity of the ship for quick rescue. It so rare for a plane to lose power to all engines though that you should never hear of such an incident. Depending on the route, however, Bermuda may be within gliding distance.