# How far would a paper plane thrown from 35000ft fly?

29 September 2015

## Question

If you were able to take a paper airplane and throw it from 35000 ft, how long will it fly and how far will it go?

We put Mo's question to Chris Smith, to do some number crunching...

Chris - A long time. Let's think about this in several ways. One is, what's the objective evidence? Well, there was something called the Space Planes Project and I actually was doing a bit of reading about this because, in 2011, there was a project to deploy a fleet of 200 paper aeroplanes from almost the verge of space. Joel Veitch, whom we've actually had on this programme, does interesting things: one of these things he decided to do was to deploy this fleet of aeroplanes using a helium balloon, which ascended to 23 miles above the Earth's surface; balloon pops; planes deployed. The planes were endowed with a memory chip so that people could find them, read what's on the memory chip, and then they could contact Joel Veitch and his team to tell them where they had found an aeroplane. It's a bit like, in the old days, you would sort of release a helium balloon. If you found the card that had, "Please write back to me. I live at." on the card then it was a way of corresponding with people before the internet and Facebook came along. They released these things 23 miles up, which is three times the height we're referring to at 35,000 feet. But they did get reports, having released them from Germany, of planes being recovered across the northern hemisphere: Canada, America, they've got some planes back. The best figures I've seen for the performance of a paper aeroplane is that for every about 12 metres it goes along, it drops 1 metre. But the fact is, if they did pick some up in America, it flew for many, many hours to have made it that far from a German launch point. That's the first point. The second point is, what's the absolute minimum we can put on this? How long would your plane be expected to stay airborne at a minimum? Well say, you are really, really unaccomplished at paper aeroplane making and you manage to fashion a paper ball.

Kat - That's my level.

Chris - Well then we can use a simple bit of physics to work out how long that would stay airborne for. You need one of these equations of motion. So let's say the equation is d (distance) equals - you do a half times acceleration due to gravity times time (t) squared. So we need to do a bit of rearrangement. So, the distance is 35,000 feet - so that's about 11,000 metres. 11,000 metres equals a half times the acceleration due to gravity: that's 10 times t squared. So we're going to times by two across the board to get rid of the half. That's 22,000 metres equals the acceleration, which is 10 times the speed of gravity, multiplied by the time squared. Now, we're going to divide everything by 10 because the gravity is about 10 metres per second per second. So, we've got about 2,200 equals t squared; the square root of 2,200 is about 50. So, your ball of paper would fall for at least 50 seconds from 35,000 feet. In other words, you'd achieve, even if you'd made the world's worst paper aeroplane, a flight time of a minimum of about a minute.

Andrew- It probably wouldn't actually reach terminal velocity!

Chris - That's not including air resistance I should say!

Andrew - I suspect it would last a lot longer than that!