Holding The Mirror Up To Nature: Critique of Mimesis of Flash Jump

Purpose:
Exploring mathematical accuracy in relation to reality of Hermit skill, Flash Jump (hereafter referred to as FJ), on different fronts:
a) Height-to-jump ratio
b) Length of jump
- Ratio consideration
- Energy conservation
c) Kinematics

Materials:
1. Test subject would be my Level 92 Hermit: Blushes, with speed 140% and jump 121%, under the effects of Haste
2. Max-levelled (20) Flash Jump
3. Snip & Sketch functions
4. 50 mesos
5. iPhone stopwatch
6. 4 hours of my time

A) HEIGHT-TO-JUMP RATIO
Measurements were recorded and referenced against the in-built ruler in Snip & Sketch.

Figure 1: Respective Measurements of Each Variable​

The data acquired shows that Blushes is 15 units, while the distance from the ground to her feet at her maximum height is 24 units. For simplicity sake, the ratio is 5:8.

The current world record for high-jump is held at 2.45m (https://www.worldathletics.org/records/by-discipline/jumps/high-jump/outdoor/men), and assuming Blushes is somehow an Olympic Champion, Blushes would have to be taller than 1.53m to achieve the 5:8 ratio. Based on global height data (https://ourworldindata.org/human-height), the approximate height of 1.53m is very realistic. The assumption here is that 2.45m was recorded from the ground to the base of her feet, as per Olympic guidelines, instead of the ground to Blushes' Centre of Gravity.

While there may be a variance in Jump % under the effects of Haste in different characters, which may result in deviating datapoints being collected, my objective is merely to prove the plausibility, not the probability. As such, the height-to-jump ratio of FJs portrayed in MapleLegends is indeed replicable on a human-scale.

Error Analysis:

1. I dropped a coin to mark the starting point, and I try to align Blushes' x-midpoint with the coin's x-midpoint to the best of my abilities. Due to discrepancies on x and y location the moment flash jump is triggered, the resulting apex of each flash jump trajectory, and similarly, x-end point, will differ.
   
   Therefore, this experiment was repeated 5 times, as shown by the 5 coins placed. Each jump was video recorded, sifting through each frame to find the perceived apex. Frames depicting the apex of the 5 different jumps are subsequently compared with each other, to make sure that the differences lie within a 5% error margin, therefore confirming the accuracy of the results. Meanwhile, for the x-axis (coin drops), the centre-point of the median coin is assumed.
   
   
2. Blushes' 'ahoge' will not be taken into her height's account for greater verisimilitude, and to also have a closer resemblance to the average character.

B) LENGTH OF JUMP
With reference to Figure 1, FJ covered 80 units of horizontal ground. This would put the height of Blushes, 15 units, in a 1:5.33 ratio to the x-distance covered.

Using the same ratio-method of analysis as Section a), with the broad jump world record being 8.95m (https://www.britannica.com/sports/long-jump), then Blushes must be approximately 1.68m, which is a very reasonable height, as corroborated by an aforementioned reference.

An important issue I would like to highlight about the ratio-method is that it should be taken merely as a ballpark figure; it is imperative not to conflate 'ability to jump' with 'height'. This simply means that you do not have to be exactly >1.68m to achieve the goal, like how not every >1.68m person can jump 8.95m. The intention of ratio method is only to check whether the values are realistic - for example, it cannot be that height:FJ is in a 1:3 ratio, otherwise Blushes would have to be 3m tall.

In conclusion, according to ratio-method, x-distance that FJ covers is realistic.

Given that y-distance was previously proven to be realistic, and now x-distance is also proven to be realistic, I see no reason why an amalgamation of both x & y would alter that (double positives = positive). Just to be sure, the y:x ratio in game is 3.33:1 (simplified from 80:24), while the world-records are 3.65:1 (simplified from 8.95:2.45). Since this is a feasible approximation, then FJ proves realistic by ratio-method no matter how you spin it.

Now, the ensuing paragraph will use Energy Conservation Laws to deduce the feasibility. To construct such an argument, the following premises will be assumed:

1. Assume g = 9.81 m/s^2
2. Closed system, for application of Conservation of Energy (i.e. total internal energy of the system is constant, therefore KEsystem + PEsystem = Etotal at all times)
3. Centre of gravity of Blushes lies at 1/2 of her y-height, i.e. 7.5units

Applying the formula for Gravitational Potential Energy at the apex of her trajectory, Blushes possesses GPE of (7.5u+24u)mg. Since Kinetic Energy is 0 at the apex, this would mean that total energy in the system is 31.5umg. In other words, she must possess 31.5umg of KE the moment right before she lands. With the formula of KE being 0.5umv^2, we can come up with the equation 0.5umv^2=31.5umg. The value of velocity in purely x-direction, variable 'v', will be 24.86m/s (~90km/h). Unfortunately, this is completely improbable for running speed - with Usain Bolt holding the record at 9.58s for a 100m sprint, his average speed in m/s translates to only 10.44m/s. However, since this is a flash JUMP, it is not impossible to think of it moving at the same speed as a car, provided only one iteration of FJ is performed in real life before taking a break (unless you have superhuman stamina and can proceed to run > long jump > run > long jump).

Therefore, I personally think FJ velocity is not too far-fetched, but require more data to make an objective conclusion. Furthermore, these calculations are made under the assumption that gravitational acceleration is the same as Earth's, which is theoretically unlikely, since it requires the Maple World to have similar mass and radius as Earth (g = GM/r^2).

An alternate solution to finding gravity empirically would be to measure the time it takes for Blushes to land after she jumps. Since there is no air resistance in maple, ma=mg for both the journey off the ground, and toward the ground. Given same acceleration and distance travelled, the time to reach apex from ground and the time to reach the ground from apex is exactly identical. Following the previously assumed ratio of 24 units representing 2.85m for y-axis, 31.5 units is hence substituted by 3.75m (Blushes' C.O.G to ground).

After taking several readings, it seems that they converge towards 0.8 seconds. This is to say that travelling 3.75*2 m takes 0.8s, which puts us at an acceleration of ~9.4m/s^2, confirming that FJ's intended speed is roughly 90km/h, since gravitational fields are approximate.

As an attempt to compare running speed with Usain Bolt, I used an iPhone stopwatch to time how long it would take me to cover 80 units. If the running speed values tally, it grants more confidence to assume that FJ velocity was designed to be that fast.

I repeated this experiment 3 times, and it took me an average of 2.30 seconds (respectively 2.23, 2.37, 2.30) to cover the horizontal distance, with Speed 140%. Margin of error, due to human reaction time, is calculated as (2.37-2.23)/2.3 = 6%, which is reasonably small enough, considering how short each experiment is.

Using the previous value of 8.95m horizontal distance, this would mean that Blushes is walking at a speed of 8.95/2.3 = 3.89m/s (14km/h, 8.7mph), which is actually a reasonable running speed. To give perspective, it is completing a 400m lap in slightly below 2 minutes.

As such, given that all the other calculations are realistic thus far, I am inclined to believe that the speed of FJ was purposely designed to be as fast as a car, and in that regard remain feasible.

C) KINEMATICS
To verify the 90km/h FJ, which is currently the only debatable value, we can attempt to use kinematics.

The timing for each FJ is approximated to be 0.7s - the margin of error is high since controlling 3 keys while lapping the stopwatch proves rather intractable. However, by asking a volunteer with the same jump % to jump at the same time I FJ, it seems that the timings coincide relatively well. Recall that the previous value calculated for each jump was 0.8s.

EDIT ON 12/1/2021: I accidentally made a careless mistake of using Vx 90m/s instead of 90km/h. The calculated Vx, 25.57m/s, is approximately 90km/h (92km/h to be exact), ergo kinematics calculations are actually accurate, and the following postulations can be disregarded.

However, Vx = 25.57m/s only. To achieve Vx = 90m/s, by working backwards, t must be 0.2s.

Figure 2: Calculation of Vx Using Kinematics Laws
​

Some reasons I can postulate for having conflicting velocities:
1. Variable 't' includes the whole FJ time, including the y-component, since it is necessary to jump before the skill can be triggered. Still, no matter what, that would account for <0.4s since it precedes the apex - there is minimally another 0.2s unaccounted for.

2. Variable 't' is too short to work with, and will have high degrees of deviation no matter what, given animation lag and human reaction time.

3. 0.2s is too fast for actual gameplay, in terms of both the software and user - imagine a FJ that acts 1/4 of the current time; that would be too fast for minds to process, and consequently result in choppy gameplay.

With the time taken to FJ, 't', being so short, parred with inevitable systemic inaccuracies that pose large margins of error, this is definitely not a method, nor dataset, that I can stand by.

To conclude, taking everything into account, it seems that there is surprising verite to FJ mechanics, despite some variables being incongruous.

TLDR; holy shit maple mechanics hella realistic tbh...

EDIT ON 12/1/2021: Here are the TLDRs, (informally)
1. Realistic height-to-jump ratio (5:8)
2. Realistic length of jump ratio (1:5.33)
3. Perfect energy conservation, not kidding literally perfect speed and perfect correlated height attained, this makes me think they deadass have a physicist on their team. Like literally exact height to exact speed is correct. (Total energy of 31.5umg)
4. For x-axis, exact distance travelled (8.95m) to exact speed (25.571m/s) is correct, its actually ridiculous how accurate it is (derived from correlation between section B and C)
5. Flash jump animation timing (0.7s) is accurate to the 0.1s. How are they doing this???
6. G force is realistic, x-acceleration is 36.53m/s^2, ~3 Gforce is about the speed of a rollercoaster.
7. Max speed (aka under haste, 140%) is running speed irl (14km/h, 8.7mph), while FJ speed is similar to a car's (90km/h)

if u read up till here, congrats,,?? idk why u would do that,, but thx,,,, yes this is a shitpost,, pls feel free to hmu to contend anything in pertinence to either the premises or actual math!! i havent done energy principles in a long long time :,) and lmk if u dont understand anything cos idt im that clear in my explanations anw,, was fun to make tho!!

s/o to marc & ash for being the only people excited to read this... u guys r nerds...

 

i.. u... ur name is literally clevercookie... :~(

 

afk...

 

HI!! firstly, thanks for supporting all my threads thus far <333 and for taking the time out to read it so fastidiously!!

u r right!! i think 24.86m/s should have reflected root mean square speed instead of pure x-velocity!
however i think x-velocity wont be a constant since there is acceleration applied

these are the following changes i've revised, and in fact this reinforces section c), aka the kinematics component, because they produce a similar Vx value

in finding angle of inclination right before landing:


applying 40 degrees into calculation:


what this means is that literally every aspect of flash jump is mathematically viable in real life... im sho0k...

 

1. w... WHAT IS A .TEX TEMPLATE... i m forum noob....
2. yo i was actually about to cite using EEE but then i was like,, who,,, who is going to read it??? i didnt expect this many nerds :,)