So, my friend showed me this video on YouTube from the channel Game Theory, which claimed that Sonic, especially on the Genesis, is not as fast as we were led to believe:
I felt, however, that this video made quite a few assumptions about Sonic, though, and as one of the video's comments pointed out, perhaps time is slowed down for us so we can react quicker, and Sonic is really going near the speed of sound after all.
As a college student currently working towards a physics major myself, I decided to see if I could find a good measurement for Sonic's speed on the Genesis without using the time variable and in a way that would potentially maximize his speed, since I want to make him look as fast as possible.
The best chance I felt I had at this was from act 1 of Angel Island from Sonic 3. In the stage, Sonic runs along the inside of a tree in a display that seemed to require a lot of speed to accomplish.
Since the height given in the video for Sonic of 1.1 m seemed reasonable, I measured the tree using that. The height Sonic must ascend is about 7 Sonics tall and 6 Sonics in diameter (or 3 Sonics in radius); this means it is 7.7 m tall and 3.3 m in radius.
To run up the tree without falling, Sonic must produce a force of friction on the walls of the tree equal to the force of gravity putting him down. Additionally, he must have enough kinetic energy to climb the 7.7 m of the tree.
First, the force of friction, which we can find from the following process:
F_gravity = F_friction = μ*F_Normal
m*g = μ*m*a_centripetal
If Sonic's planet is like Earth, then g = 9.8 m/s2, and μ equals the coefficient of friction for wood, or about 0.4. The masses m cancel out. So...
9.8 = 0.4*a_centripetal
So centripetal acceleration is 24.5 m/s2. The equation for centripetal acceleration is a = v2/r. Since a = 24.5 m/s2 and r = 3.3 m, v2 = 80.9 m2/s2, and v = 9.0 m/s. This is Sonic's minimum possible speed at the top of the tree.
To find Sonic's minimum possible speed at the bottom of the tree to make this possible, we use the law of conservation of energy to give us the following equation:
E_Kinetic_i = E_Kinetic_f + E_Potential + E_friction/heat
0.5*m*v_i2 = 0.5*m*v_f2 + m*g*h + μ*m*a_centripetal*distance
m is cancelled out, v_f = 9.0 m/s, g = g = 9.8 m/s2, h = 7.7 m, μ = 0.4,and a_centripetal is 24.5 m/s2. In this equation, we are trying to look for v_i, which will give us the speed needed for Sonic to do this.
Finding distance is a little more tricky - this is the total distance over which Sonic travels as he runs around the inside of the tree. Playing through the level, Sonic makes 2.5 rotations around the inside of the tree. The formula for a circle's circumference is 2*π*r, so the total distance of 2.5 rotations is 5*π*r. With r equaling 3.3 m, this total lateral distance comes out to be 51.8 m.
The vertical distance traveled is 7.7 m, so we can use the Pythagorean theorem to find the total distance:
dist2= x2 + y2
dist2 = 2683.2 m2 + 59.3 m2
dist2 = 2742.5 m2; dist = 52.4 m
So, the conservation of energy equation now looks like:
0.5*v_i2 = 0.5*80.9 + 9.8*7.7 + 0.4*24.5*59.3
v_i2 = 2*(40.5+75.5+581.1) = 1394.2
v_i = 37.3 m/s
So, the initial speed Sonic would need to run up a tree like when he does at Angel Island in Sonic 3 would require him to run at 37.3 m/s. As you can see, I never used the time variable to determine this, so the question of whether or not time is slowed down for the player doesn't matter.
This is still much less than the 340.3 m/s needed for supersonic travel at sea level, but more than the speed given in the video for Sonic on the Sega Genesis. Now, I invite everyone else to come up with more tests that maximize Sonic's speed on the Genesis without using the time variable. I'd like to see how close we can get to justifying Sonic's claim to break the speed of sound on the Genesis using physics. The faster, the better. Good luck!