So I was watching episode 8 of Defenders of Berk this morning and I noticed something odd in the opening sequence. After a few tries, I captured a screenshot of what I noticed. It was this moment:
What we have here is a very clear example of a Mach cone streaming behind Toothless and Hiccup, meaning that they have broken the sound barrier.
So, seeing as how I learned a lot about sound waves and relationships in physics in university, I decided that I would try to calculate Toothless’ speed at this moment. I don’t know if anyone has done this before, but I spent most of today on this so here goes.
The reason I say this is Toothless’ top speed is because nowhere else in the movies or shows have we seen him form a Mach/vapour cone behind him, and breaking the sound barrier is a pretty max speed so we’ll assume then that this could be his top speed.
For those of you who don’t know what a Mach cone is, the short explanation is that its the collection of wave fronts of sound waves that have constructively interfered to form one large wavefront that trails behind the object travelling faster than the speed of sound. The cone is visible because of a sudden decrease in pressure as the amplified shockwave passes by, causing the water molecules in the air to condense. If you want a more detailed explanation, reference this wikipedia article.
The speed of the object, or the source of sound and its relationship to the dimensions of the cone can be derived through some simple geometrical analysis of the cone,
which shows that the object is moving faster than the wavefronts of the sound wave trailing behind it, causing the wavefronts to overlap along their edges in a cone shape. The angle between the cone and the central axis, we’ll call it θ, can be related to the speed of sound and the speed of the source by this equation:
Where v is the speed of sound and vs is the speed of the source. For us, this means that Toothless’ speed is equal to the speed of the source. All we need to do is find the value of the angle θ and the corrected value of the speed of sound for the conditions in which they are flying.
So, looking at our photo, we can overlay a cone outline (reference for why I drew the cone like I did) onto the vapour cone in the photo, and define a few variables.
From the wikia page for Toothless, it is said that Toothless’ full length is 10.0m, so this is a reference in our calculations.
In order to find the angle θ, because the cone is being viewed from a bit of an angle we will find it through the trigonometric relationship between the radius, r, and the height of the cone, h:
Since we have the length of Toothless, we can use that as a reference to find the dimensions of the cone.
To find the radius of the cone:
And to find the height of the cone:
By orienting the length of Toothless along each of the axes we get their length relative to Toothless’ length. By a little bit of measurement and ratio calculations (which I won’t bother to go through here because they are tedious and irrelevant) we find that
r = 9.50m
h = 8.30m
By using the tangential relationship between those two values, we find:
So, now we have one of the two values we need for the Mach calculation. The other value we need is the corrected speed of sound.
The speed of sound in dry air at 0°C is 331.3 m/s. Now, although I do know the formula for the corrected value of the speed of sound in dry air, it is not helpful here because Toothless and Hiccup are flying over water, and there is bound to be humidity affecting the speed of sound.
By looking at the image, we see a bit of mist formed around the surface of the water, (despite the poor quality of the image but I am sure that there is some mist there).
Using a reference from the UK, for aviation purposes, a visibility of less than 5 km but greater than 999 m is considered to be mist if the relative humidity is 70% or greater. Thus we can assume that the relative humidity over water, because there is some mist, is at least 70%.
The other thing we need in order to find the correction for the speed of sound is the temperature. By looking at a world map of average temperatures throughout the year, and knowing the historic areas where vikings lived, we can ascertain that the temperature is fairly cool. There is no ice on the water, so we can safely say that the temperature is above 0°C. Considering there is some considerate cloud formation and that the mist formed over water usually forms at relatively cool temperatures, and because Berk is supposed to be a very cold place, let’s ballpark the temperature to be about 10°C. Not very warm, but not too cold either.
Taking these two estimates, we can calculate the corrected value for the speed of sound. Since I was unable to find the formula for this, I just plugged it into a program that I came upon and it spit out a number for me. (Note that the air pressure at sea level is 101.325 kPa)
This gives us the speed of sound to be 337.9 m/s.
Using the speed of sound and our Mach cone angle, we can finally calculate Toothless’ speed:
Toothless and Hiccup’s speed in this shot is given to be 448.4 m/s, or 1614 km/h.
Now, that seems like a pretty high speed for a boy and his dragon to be travelling, so I looked up a few references for how fast this is compared to say, modern day airplanes and whether or not Hiccup would survive travelling at that speed.
Now, the average speed of a passenger jet plane is 680-1000km/h. There are other jet planes that could go up to a speed of 3000km/h and above. The fastest powered jet plane is the SR-71 Blackbird with a speed of 3230km/h. So Toothless is travelling faster than a passenger jet, but still within an achievable range of a faster jet plane. Some of the fastest jet planes travel at Mach 5, but Toothless here is travelling at a modest Mach 1.3, or 1.3 times the speed of sound. So this speed is relatively reasonable. Considering the huge minimum flight speed Toothless needs in order to stay aloft, (at least 100 km/h, referencing from someone else’s analysis), this should be perfectly attainable for him.
As for Hiccup, this speed is definitely a large speed for him to be able to survive. I’m not going to get into the fact that he is wearing next to no protection and would probably be blown off by the sheer force of the wind he’s facing. The question here is: would he live?
Humans have broken the sound barrier before, but they have always been in the confined protection of the cockpit. Our real life reference needs to be of someone who was facing the elements without a metal shield to protect him. So, let us considering Felix Baumgartner’s incredible free fall. Recently, Felix Baumgartner became the first skydiver to go faster than the speed of sound, reaching a maximum velocity of 1,342km/h.
So, Hiccup and Toothless are travelling at 1614 km/h, which is still faster than Felix Baumgartner did in his free fall from the edge of space. However, judging by Felix Baumgartner’s survival, Hiccup would probably survive this flight speed considering that there are no sudden stops awaiting him, but rather Toothless will control the acceleration so Hiccup shouldn’t be dead from impact force.
One interesting side effect of travelling this speed however, will be that Toothless and Hiccup will not hear anything. Because they are travelling faster than the speed of sound, they are moving faster than the wavefront of the sound waves can reach them, so they are flying into a remarkable silence. This could be dangerous for Toothless and his navigation skills since we know that he can use echolocation for navigation, but considering this stunt was done in an open space in daylight, they should be just fine. In any other situation though, I would be skeptical.
So there you have it folks, Toothless’ max speed calculation based on the appearance of a Mach cone in the opening reel of DoB! If you made it to the end, thanks for reading this and I hope it was interesting!
TL; DR
Toothless can fly at speeds of up to 448 m/s!
turnnoffyourmind
annaxmalina
romaincrisis
titsay
