Once in a blue moon, there is a man gifted with the ability to nitpick any game he plays. This man is named Austin Hourigan.
The Game Theorists channel has done many Sonic episodes in the past, including the theories concerning the Chaos Emeralds, Sonic’s speed, and Tails. Because of this, Hourigan has been a bit hesitant about making a video on the franchise. But after playing Sonic Mania, he has had an urge to discuss this game or more specifically the famous loop in the Green Hill Zone.
He begins by comparing the loop with the Halo Ring from the game Halo and a rollercoaster. In the Halo Ring, though you appear to be stationary, you are actually moving along with the rotation of the ring. The energy of the gravity is already in the rotating ring, unlike rollercoasters where you are putting the energy as you enter the loop. This is the case with the loops in Sonic. If you go fast enough and redirect the energy, you can make it through a loop.
The problem with Sonic, however, is that he is using his feet which doesn’t redirect energy as efficiently as wheels.
In order to calculate how fast Sonic needs to go in order to go through the loop, Hourigan uses the formula, velocity equals the square root of gravity divided by the radius of the loop. To find the gravity, he uses a Japanese site that details Sonic’s height which is a meter. Using Sonic’s height to measure how quickly he falls thanks to gravity, he concludes that the acceleration of gravity is 21.43 M/S² (meters per second squared) which is twice as much as the Earth’s gravity. After calculating the radius of the loop using Sonic’s height, he plugs the numbers which turn to 6.6 M/S or 14.76 MPH (miles per hour) in order for Sonic to go through the loop which is, admittedly, pretty slow.
After briefly mentioning that Sonic’s acceleration is 3.47 M/S², Hourigan shows how the game somehow lets Sonic go through the loop despite only going 5.3 M/S. This leads to the suggestion that Sonic may be gaining speed as he is going up the loop. He responds to this notion by pointing out that Sonic will be fighting gravity more as he is going through the loop. At the 90 degree point, where the acceleration of gravity cancels out Sonic’s, he will slow down. He will also accelerate less as he reaches that point. Our scientist then calculates what angle gravity needs to be to cancel out the blue blur’s acceleration of 3.47 M/S². He begins by finding the B Vector which is the square root of Sonic’s acceleration (3.47 M/S²) squared plus the acceleration of gravity (21.43 M/S²) squared which equals 21.15 M/S². He then divides that by the gravity vector and takes the arctangent of that answer which becomes 44 degrees (technically if you round it up it is 45 degrees but minor nitpick). He then subtracts 180 degrees with 44 degrees which becomes 135 degrees.
This means that once Sonic reaches 18 percent of the way through the loop, all his forward acceleration is canceled. This means that he will only reach a top speed of 5.8 M/S. In other words, he would not be able to gain enough speed as he is going through the loop. This is, however only his slowest possible speed. Hourigan moves on to Sonic’s fastest speed while running which is 10.3 M/S. So you would think he is fast enough to go through the loop right? WRONG! There is another thing to take into account, energy balance.
When Sonic reaches the top of the loop, his energy will be less than or equal to the energy Sonic had when entering. When entering, his energy is all kinetic. But as he is going up the loop, it becomes potential energy. At the top of the loop, kinetic energy is at is lowest and the potential energy is at its highest. The kinetic energy starts to rise again as Sonic leaves the loop. In order to find how much energy Sonic needs to do this loop, he finds the kinetic energy which is Sonic’s weight (35 kg) multiplied by the minimum speed to go through the loop (6.6 M/S²) squared. The answer is then halved which is 762.31 joules. Hourigan then finds the potential energy which is mass times gravity times height which becomes 3049.24 joules. After adding the kinetic and potential energies, he concludes that Sonic would need 3811.56 joules of energy to go through the loop. By rearranging the kinetic formula to have velocity on the left side, he plugs the numbers to get 14.75 M/S or 33 MPH. This is how fast Sonic needs to go to provide this energy.
So, can Sonic make it through the loop? Well, not by running apparently. But Hourigan then moves on to Sonic’s ball form. When Sonic is fully revved up, he goes 16.57 M/S meaning he can go through the loop.
So, what did you think of this video? Comment below on your thoughts.