High altitude operations

This topic is usually asked in a checkride by examiners. Let us try to understand what questions are asked and how to respond to those.
Before that I would like all of you to read my previous post about relationship between pressure, temperature, density and altitude. 
Effect-of-temperature-and-pressure


Also see the following mathematical relationship:

where

• L is lift force,
• ρ is air density
• v is true airspeed,
• A is planform area, and
•  is the lift coefficient at the desired angle of attack, Mach number, and Reynolds number

In summary, At high altitude, you have lower pressure, lower temperature and hence lower density (i.e. amount of air molecules per cubic meter). Now the lower density is the main reason which affects the takeoff/landing. Let's see how.

Question1: How is takeoff affected by high altitude?

1. Less engine performance -- Engine requires air to operate (i.e. burn fuel with air). But at this altitude density is less and hence less air for engine. This results in lower performance of the engine. That is one reason why you may (e.g. in cessna 172 its recommended to lean mixture above 3000ft) need to lean the mixture to correct the air/fuel mixture for best performance.
2. Higher TAS -- Higher true air speed. TAS is relative to the air your are currently travelling. Since engine performance is less, you will need to travel faster to get same amount of air required by engine for best performance. Note your indicated airspeed will remain the same i.e. you will still takeoff at the same indicated a/s.
3. Longer takeoff roll -- Since engine performance is low, aircraft needs to travel longer and faster to get the required amount of air to burn fuel optimally. Imagine you are travelling in a car and you open the window and keep your hands out. At low speed you will not feel the amount of air pushing your hands but as speed increases you will feel more air pushing your hands. Its the same principle. As you travel faster in aircraft, more air will start hitting your engine and it will be able to generate higher thrust. As part of flight planning you should calculate the takeoff roll so that its within the runway length. 
4. Reduced maximum takeoff weight -- Higher weight may result in even longer roll as aircraft needs to travel faster to get the amount of lift to balance that weight and runway may not be long enough to achieve that required speed.
5. Reduced Lift -- As you can see from the mathematical relationship above, density is directly proportional to lift. Low density translate to less lift.

Question2: How is landing affected by high altitude?

1. Higher TAS -- As above, since amount of air is less to reach the same indicated airspeed required for landing the TAS is high. 
2. Longer landing roll -- As TAS is higher the aircraft is travelling faster. To slow down it will need longer runway to stop. Also, since air resistance is low that as well contributes to longer landing roll.
3. Less resistance to land -- As air is thin, it provides less resistance for the aircraft to stop and increases landing distance.

These are some of the reasons, why performance is always calculated against density altitude. Note the above reasons also apply to airport with high temperature, as that also results in lower density. Hence if you are at sea level but very hot for the day and lower density, it will behave like high altitude airport.