Keanu's falling arm conjecture

Suppose we put a theoretical humanoid robot on solid ground. The robot has a detachable arm from the shoulder down. Suppose we tell the robot to find a way to detach its arm while allowing it to fall to the ground without any of its components, resulting in any damage to the arm's integrity. The arm of the robot has a special function that allows it to extend its arm from the elbow down. Is it likely that by extending its arm to a greater length, it may fall to the ground without any detrimental impact towards the delicate components? By gaining an increase in arm length, the hand is then supported by this increase in length, in terms of slowing down the fall. (Which is the part of the falling mass that interacts with the ground first) This means that the overall height of the fall decreases along with the pull of gravity aimed toward the arm which then ultimately weakens the final force of the arm landing on the ground. By optimizing the length of the arm, we are able to safely land the arm onto the ground.Think of the volume of the arm as a rectangular prism, or even a cylinder. we think of the length of the arm as the height of the cylinder. If you imagine it as a rectangular prism, on the other hand, think of it as the width of the rectangular prism. We then focus on that height/width of that arm, as it then tells us how far apart the hand is from the elbow. This then allows the space for more possibilities in the amount of force that is being generated as the arm falls to the ground.

M = Mass of the arm

V = Volume of arm

W = Width

H = Height

G = Gravity

D = Distance

F = Force

I = Impulse

t = time

UM = unit of measurement

End force of arm colliding with ground:

t=GD=9.8ms2UMF=M×GΔP=F×t

Decrease in height/distance from the arm to the ground:

Cylinder:

if H+ΔH then D−ΔH

Rectangular prism:

if W+ΔW then D−ΔW

When doing the experiment, I included struts to support the legs that are planted into the ground. This was done to minimize any recoil force when the arm decouples from the body. As the detachment function uses a slight jolt of force from the pressure that releases from the arm separated from the rest of the body.

Along with there being pre-set angles for the elbow to minimize the distance from the ground to the harm. The angles of the elbow is as follows:

Angle of elbow bent inward: 20.5°

Angle of elbow swiveled inward: 81.7°

The elbow of the arm was set at these angles to minimize ground impact from a height perspective. As the hand will be impacting the ground first. It also is crucial for the arm to land on the hand so that the arm doesn't fall on to its sides.

angle of the elbow in relation to the ground

How will this be tested?

A ratio between sucsessful and unsucsesful landings will be used.
The ratio is set to the limit of 10, meaning that the amount of successes and failures must equal to 10.
This ratio is then used for a probability simulation.
The probability tells us how likely the arm is to safely land on the ground.

Results:

P(arm landing on the ground safely) = 60%

M(average of sucsessful landings) = 2.53

Based off my simulation, the probability of the arm landing on the ground without any of its components being broken is 60%

This means that when doing the trials, most of the iterations within the trials resulted in a sucsessful landing.

The average amount of iterations a sucsessful landing shows up in is within approximately 3 iterations. This further supports the findings of how likely it is estimated to land safely.

This simulation is without a doubt not completely accurate. As this was only a estimation of how the statistics of the outcome should look like. If I were to restart my simulation of 30 trials, I am most certain that I would of gotten diferent answers.