Sonic (The Hedgehog) Speed
Sonic is a blue hedgehog from a series of SEGA video games (dating back to just after the Mario games came out) that collects rings and goes fast. In fact, going fast is the premise of most of the Sonic the Hedgehog games, with the primary goal of each level being getting from the beginning to the end as quickly as possible (and hopefully looking rad on the way). One of the level elements that appears many a time is a loop-the-loop, both to demonstrate how fast Sonic is moving, and to be a barrier which cannot be surmounted unless Sonic is moving at the necessary high velocities. So, this brings us to the question: how fast does Sonic need to go to complete a loop-the-loop without falling?
Well, the basic principle of a loop-the-loop relies on the idea of centrifugal, or, more accurately, centripital force. This is a fictional force that appears when you're in a rotating system. If you attach a rock to the end of a length of string and begin swinging it in circles, the string will eventually become taut as the rock appears to be pulled outward away from the center of rotation. This apparent force is called "centrifugal," or "center-fleeing" force. The reason I say "apparent" is because the force isn't real. The rock feels what seems like a force pulling it away (and if you've ever ridden an amusement park ride that puts you in something and spins rapidly, you also can feel the apparent force holding you to the wall), but that force is really a product of a continuous acceleration as you rapidly change direction. The only force that you would see were you not in the rotating reference frame is the force on the string keeping the rock from flying away in a straight line (that is, the "centripetal," or "center-seeking" force). These so-called fictional forces arise whenever you are in a non-inertial reference frame, which is a fancy way of saying that you're accelerating. As I covered in The Physics of The Flash, acceleration is both a change in speed or a change in direction. So, when you're in a rotating system, you're constantly accelerating, which puts you in a non-intertial reference frame. So, as a product of that acceleration, you perceive a force that wants to push you away from the center of rotation.
This center-fleeing fictional force means that there's a force that is going against the gravity that wants to pull you down at the top of the loop. This means that the minimum speed you would need to complete a loop-the-loop would be a speed which puts your centrifugal force exactly equal to the gravitational force at the top of the loop. But really, in order to fall, what we care about is acceleration. If you have no downward acceleration, you're not going to start falling (unless you are already falling, in which case, you will continue to). What this means is that we just need our centripetal acceleration to be equal to the acceleration due to gravity. The cool bit about this is that acceleration doesn't care at all about mass. Forces do, but acceleration doesn't (that's why, in a vacuum, with no air resistance, objects of different masses/weights fall at exactly the same speed). So that means it doesn't matter how massive Sonic is, only that he achieves the necessary acceleration to make it around the loop.
So what would that acceleration be? It turns out this problem is super easy once you know what you're trying to solve (that centripetal acceleration is equal to gravitational acceleration). The equation for centripetal acceleration in a circle with a radius r is
Where v is Sonic's velocity. So, if we set this equal to acceleration due to gravity g, we get
And solving for v
So now, if we know the radius of our loop, we can see how fast Sonic has to be going to do the loop-the-loop without falling on his face. This is where things get trickier. I dug out my old copy of Sonic Advance (for the Game Boy Advance), and spent some time in the first level looking at the loops. I estimate the height of a loop-the-loop to be about five times the height of sonic. That would put the radius of said loop to be 2.5 Sonics. Now, we could solve the equation in Sonics per second, but that's silly, so instead I can look up Sonic's official height, which is evidently one meter (opposed to the height of a typical hedgehog, which is a quarter of that at about 25 centimeters). That means our Sonics per second unit is actually meters per second, which is a wonderful crazy random happenstance. That puts our radius at 2.5 meters, which means that, solving for v (with g = 9.8 m/s2), we get a necessary minimum velocity of 4.95 meters per second, which is only about 11 miles per hour. Which is disappointingly slow.
Nevertheless, let's see if Sonic moves at sufficient speeds to make the loop-the-loop. Again in the first stage (Neo Green Hill Zone) of Sonic Advance, the slowest speed I'm able to make the loop-the-loop at is in a little under two seconds. Let's be generous and call it 1.5 seconds. The circumference of that loop would be about 15 meters (a bit over, five meters times pi), which would put Sonic's velocity at 10 m/s (22 mph), or twice what he needs to successfully complete the loop-the-loop. So, is Sonic Advance a scientifically accurate game? Well, the biology is a little iffy (a one-meter tall blue hedgehog), but from a physics standpoint, it's A-OK.