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Sonic's Top Speed on the Genesis - The Physics of Sonic


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#1 Mesmerist

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Posted 03 July 2013 - 02:39 PM

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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:

http://www.youtube.c...h?v=ze_uxxFJTvE

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!



#2 Mesmerist

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Posted 03 July 2013 - 02:58 PM

 

 

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

 

Whoops! I used the wrong distance in the final equation! Let's try that again:

0.5*v_i2 = 0.5*80.9 + 9.8*7.7 + 0.4*24.5*52.4

v_i2 = 2*(40.5+75.5+513.5) = 1259.0 

v_i = 35.5 m/s
 

This is the same as 79.4 mph or 127.8 km/h. The maximum speed of a cheetah is 120 km/h (from Wikipedia, but heavily backed up by citation). So, even if Sonic isn't capable of supersonic speed based on my calculations, by the time Sonic 3 came out, Sonic really was the fastest thing alive (on land)! Still, I look forward to seeing how fast you can get Sonic to go!



#3 Jolt_TH

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Posted 03 July 2013 - 03:02 PM

Although I flunked Physics I'm glad I understood most of that, shows 2 years wasn't in vain!!

Would a good test be to try and determine his speed on a loop?
He would have to go upside down and we have sonics weight and height, surely we could measure the diameter of the loop using "sonics"?

#4 Mesmerist

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Posted 03 July 2013 - 03:12 PM

I tried to do calculations on the Green Hill Zone loop and came up with something uninspiring like 8 m/s... 
However, if you want to use a different loop, look for a very large one. The higher and higher you go, the more potential energy you have, so you need more kinetic energy and therefore more speed at the bottom to do it.

Also, like I said, for circular motion, a = v2/r, and at the top of the loop, a will always equal g, or 9.8 m/s2.

so, the speed at the top of the loop equal the square root of the radius and 9.8 m/s2. The greater the radius, the greater the speed must be. 

Overall, then, if you want to use a loop, use a very big and especially a very tall loop to do it.

You can also try looking for a maximum speed on the 3D games if you can outdo the Game Theory figure of 80 m/s. Again, good luck!



#5 Jolt_TH

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Posted 03 July 2013 - 06:30 PM

The last thing would be to determine how fast he is moving on order to run into of the water in Hydrocity?

#6 Mesmerist

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Posted 04 July 2013 - 12:02 AM

The ability to run on water has less to do with how fast you're traveling and more to do with how much force per area Sonic's feet press down on the water. He has to have quick feet to achieve this, therefore. At 20 degrees Celsius, the surface tension of water is 0.0728 N/m, if that's any help.



#7 Jolt_TH

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Posted 04 July 2013 - 08:31 AM

Having to have quick feet by extension would show how fast his feet was moving, maybe we could work out from there the speed of his movement?
Like using the distance of his stride would be the distance he moved per footstep, and if he has to say for example move his feet 100 times a second to stay on top of the water then we would times that by the distance of his stride, say 0.5m, to get the minimum speed? In the example case this would be 50m/s?

Of course that was all rough but would that be along the right lines?

#8 TailsTellsTales

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Posted 04 July 2013 - 09:19 AM

Super Sonic or does he have to be in regular form and on the ground? Because was thinking what about the Doomsday zone than. If not try adding the speed shoes. Try putting Sonic in Mario world and multiply how many times faster he is than Mario. Okay Good luck frying mind in!

Edited by TailsTellsTales, 04 July 2013 - 09:28 AM.


#9 Mesmerist

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Posted 05 July 2013 - 03:33 AM

I didn't think about measuring Sonic's stride; that's a really good idea! It will allow us to relate the speed of Sonic's feet (which is important for surface tension) to speed (which, of course, is what we're trying to determine).

So yeah, Jolt, all of that sounds like it would work really well! I'm a little too busy at the moment to try it myself, so go ahead and see if you can do it. If you're busy, too, or just don't want to do it, I can try tackling it in about a week or so.



#10 Jolt_TH

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Posted 05 July 2013 - 07:37 AM

I'll give it a go, but I only up to did A-level physics :P

#11 Ravenfreak

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Posted 09 July 2013 - 08:09 PM

Physics aside (yeah I never made it that far in HS... I suck at math...) from a programming point of view the only reason why Sonic wasn't as fast in the first Sonic game is due to a speed cap that was programmed in. Yuji Naka would get motion sickness as the rumor goes, so he is probably the one that programmed that into the first two games. (The second game got rid of it when Sonic's on the ground, but it's still present while he's in the air...) In Sonic 3 they completely got rid of that "feature". The guy who uploaded that video probably didn't know or did know but failed to mention that little fact from the first two Sonic games. Still it's great to see the amount of knowledge you have put behind this theory. smile.png




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