While modern aircraft are capable of flying over, or even through, hurricanes, safety risks remain, and carriers usually halt operations are the affected airports instead.
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Crosswinds greater than 50-60 km/h may cause airlines to delay or cancel flights. Even lower gusts can trigger flight delays or cancellations if the runways are wet or icy, as stiff winds can reduce a plane's ability to brake on the runway.
Flights may be canceled due to wind, precipitation, fog or low visibility, lightning, low clouds, or storms. Outside temperature does not always affect flights. Pilots make the final determination as to whether a flight will occur or whether one in progress will be diverted to another airport.
With this in mind, horizontal winds (also known as “crosswinds”) in excess of 30-35 kts (about 34-40 mph) are generally prohibitive of take-off and landing. As far as how this happens, it depends on where you are in flight.
There is no single maximum wind speed that cancels flights, as it depends on the direction of wind and phase of flight. A crosswind above about 40mph and a tailwind above 10mph can start to cause problems and stop commercial jets from taking off and landing.
Rain is just water, no matter the pressure. Modern aircraft can generate lift regardless of the heaviness of the rain. Planes can and will take off and land in the rain. The only real problem with heavy rainfall is the decrease in visibility for the pilots.
There is no single maximum wind limit as it depends on the direction of wind and phase of flight. A crosswind above about 40mph and tailwind above 10mph can start to cause problems and stop commercial jets taking off and landing. It can sometimes be too windy to take-off or land.
For example, the Boeing 737 has a maximum crosswind landing factor of 30 knots (just over 55 mph). So that answers the second part of your question. Yes, a 'plane can land in 50 mph winds.
While high winds (a crosswind above 40 mph and a tailwind above 10 mph) can occasionally prevent planes from taking off or landing on time, winds won't put your flight in any danger.
Absolutely. Pretty much any airplane can easily fly in 25 mph winds. If that is a direct crosswind that might make landing smaller aircraft more challenging, but airliners, no problems.