Monday, July 2, 2007

Freestyle Motocross Backflip Limit

Freestyle Motocross superstar Travis Pastrana hails from Annapolis, MD. That's just down the road from my home town. Besides the fact that FMX is simply cool, having one of the most famous competitors based so nearby means that my son can't help but love watching the competitions (OK, I'll admit it, I like them too).

One day as we were watching Travis throw down backflips and double backflips on Fuel TV, it made me wonder just how many flips were at least theoretically possible. Well I finally got around to doing the calculations.

I'm not going to keep you in suspense - according to my caculations (I love saying that) the most backflips that a person could possibly do on an FMX bike seems to be 4.

This may be hard to believe, considering that nobody has managed more than a double backflip yet. In case you're wondering how I figured it out, read on for a brief sketch. If you don't really care about the physics, you can still rest assured that FMX has some room to progress, at least in the backflip department.

FMX Backflip Calculation

The first thing to consider in calculating the maximum number of backflips in FMX is that there is a limited amount of energy available. You can estimate the maximum amount of energy from the record FMX step up height (the equivalent of high jumping on FMX bikes), which currently stands at about 11 meters.

The energy necessary to lift a bike and rider 11 meters off the ground is

E=m g h= 16500 joules

where m is the total mass of the bike plus rider, which I estimated to be 150 kilograms

g is the acceleration of gravity, or 9.8 meters per second squared

and h is the height

When a rider flips, some of the energy that they would have used to achieve height is put instead into the energy of their rotation. Optimally, the energy should be split equally between the upward motion and the rotation. I can prove this, but it requires a little bit of calculus, which is hard to show in the rudimentary editor that Blogger provides. But it seems pretty sensible, if you think about it.

If anyone wants to know the details, drop me an email and I'll explain it. For now, you'll have to take it on faith.

Once you know the amount of energy that goes into getting altitude (16500/2 joules), you can figure out how high they fly during a max backflip attempt and how long they are in the air during a jump. Because they’re diverting half the total energy into spinning, and half of the total energy to getting height, they should only travel about half as high as in a record setting step up competition, or 5.5 meters (18 feet) off the ground.

To figure out how long the whole trip takes, you have to use the equation

h=1/2(g * t^2)

where ^2 means time (t) is squared.

This is the equation for how far something would travel (h) under constant acceleration (in this case, the acceleration due to gravity g) in a certain amount of time (t).

Because we know h and g, but not t, you have to rearrange this equation to find out how long it takes to go from the ground to a height h.


(^1/2 means the square root of (2*h/g))

But don't forget that the rider has to come back down as well, so the round trip is twice as long as it takes to get to h.

You can also calculate how fast they are rotating by using the equation for rotational energy.

=1/2(I * w^2)

Where E again is 16500/2 joules, I is the rotational moment of intertia, and w is the rotational speed.

A little mathematical manipulation leads to the equation

total rotations (in radians) = w*t =(v^2/g)(m/I)^1/2

I also guessed that the bike and rider's rotational inertia (I) is about 150 kilogram meters squared. It's a reasonable estimate, and I can explain it to the very curious folks, but it doesn't make much difference in the final calculation (because we're taking the square root of I).

If you plug in the numbers, you find that an FMX rider could theoretically rotate through no more than about 22 radians. Radians are the measure of rotation in the units I'm using (SI).

There are 2*pi radians in a circle, so in other words they will do 3.5 full rotations, which corresponds to a quad backflip.

Here's how 3.5 rotations equals 4 flips in FMX.

In performing a single flip - the rider leaves the hill with their front tire pointing toward the sky and comes down with it pointed to the ground. Here's a video to prove it

That means the bike and rider actually rotate only a little more than halfway around when doing a single backflip. For every additional backflip, they add another full rotation. So, 3.5 rotations is the same as a quad-backflip.

There you have it. Based on data from the record setting step up competition, we can expect, someday, to see a very daring and talented FMX rider to pull of a quad backflip, but never any more (unless they try it on a much more powerful bike).