The 2016 biographical film Sully is about the incredible true life crash landing by US Airways Flight 1549 on the Hudson River in New York, where all 155 passengers and crew survived without any major injuries. Here’s a memorable scene recreating the landing:
Math Scenario 1 – Which Airport Can You Make?
The best possible glide ratio of an Airbus A320 is 17 to 1. That means with the engine out, with conditions optimal, the aircraft can move forward 17 units of distance for every 1 unit of distance it falls – which is pretty good considering the plane is meant to only ever fly powered.
Using the concept of glide ratio, we can work out whether, without power, it’s even possible for a pilot to reach an airport before the plane crashes.
In a crash situation, the plane is often not operating under ideal “glide” conditions – so let’s assume the glide ratio on a damaged A320 might be more like 10. Let’s assume this plane is at 2800 feet when it hits the birds – just like the real A320 that eventually landed on the Hudson River.
There are two airports in the picture above. Airport 1 is directly in the current flight path of the plane. Airport 2 is off to the side, and will require the plane to turn, using up valuable altitude in doing so.
Can the pilot make Airport 1?
First we need to do some conversion to get everything in the same units:
Altitude in metres = 2800 ft * 0.3 m/ft
Altitude in metres = 840 m
The pilot can theoretically reach airport 1 if the glide ratio of 10 enables them to cover the horizontal distance (10 km) before they hit the ground.
Altitude loss travelling 10 km over the ground = distance / glide ratio
Altitude loss travelling 10 km over the ground = 10 km / 10
Altitude loss travelling 10 km over the ground = 1 km
1 km > 840 m !!!
Unfortunately the plane will lose 1 km of altitude when travelling 10 km – which is more altitude than it has. So Airport 1 isn’t reachable.
Can the pilot make Airport 2?
To make it to airport 2, the pilot will need to make a 90 degree turn and then follow a straight line to the airport. Let’s assume that after the pilot makes the turn, they will still have 7 km to get to the airport.
First we need to work out the radius of the quarter circle turn.
The formula for turn radius for a plane is:
Theta is the banking angle, g is gravity, and v is the speed over the ground of the plane
The real aircraft was going about 350 km/hr when it was hit, so let’s use that amount. Let’s also use a bank angle of 30 degrees, more than usual but possible with the aircraft. We need to make sure all the units are consistently in metres and seconds:
So we have a distance of one quarter of a circle with radius 1669 metres to travel, followed by another 7000 m.
Total distance = 0.25 * PI * d + 7000
Total distance = 0.25 * 3.1415 * 2 * 1669 + 7000
Total distance = 9621 metres
Can the pilot make it? We can calculate the altitude drop over this distance:
Altitude loss travelling 9.621 km over the ground = distance / glide ratio
Altitude loss travelling 9.621 km over the ground = 9.621 km / 10
Altitude loss travelling 9.621 km over the ground = 0.9621 km
962 m > 840 m !!!
Looks like Airport 2 is not reachable either – it’s a better bet than airport 1 but the calculations show that the plane would still crash before reaching the airport.
Math Scenario 2 – Dumping Fuel
If a plane needs to make a forced landing, they sometimes fly around dumping fuel to reduce the weight of the aircraft before a possibly hazardous landing.
If you’re a pilot in a Boeing 747 that is going to make a landing without landing gear, you might choose to dump fuel.
Let’s say you have 47 tonnes of fuel left in your tanks, and the plane can dump fuel at a rate of 1.5 tonnes per minute. Let’s also say you’re losing altitude at a rate of 0.3 km / minute and you’re currently at an altitude of 11.5 km.
Can you jettison all the fuel before you land?
Time to dump fuel = tonnes of fuel / tonnes dumped per minute
Time to dump fuel = 47 / 1.5
Time to dump fuel = 31.33 minutes
You’re also losing altitude at a rate of 0.3 km/hr. We can either calculate how long you have before you have to land, or you can work out how much altitude you’ll lose in 31.33 minutes. Let’s do the second one:
Altitude lost in 31.33 minutes = time * altitude loss rate
Altitude lost in 31.33 minutes = 31.33 * 0.3
Altitude lost in 31.33 minutes = 9.4 km
9.4 km < 11.5 km… phew!
Looks like you’ll still have about 2 km to spare in altitude after you’ve dumped all of your fuel.
Math Scenario 3 – Firefighters to the Rescue
In case your plane does crash badly, the nearest firefighters have been called to the airport you’re targeting in scenario 2. They take 5 minutes to get ready, and then can drive at an average speed of 30 km/hr to reach the airport (there’s a lot of traffic).
Can the firefighters travel the 10 km to the airport from their base and reach it in time for your crash landing?
Time left to drive = 31.33 – 5
Time left to drive = 26.33 minutes
Distance travelled = time duration * average speed
Distance travelled = 26.33 minutes * 30 km / hr / (60 min / hr)
Distance travelled = 13.17 km
13.17 km > 10 km
Looks like the firefighters will make it to the airport in time to help you in the aftermath of the landing.