Electromagnetic Soft Docking: Derivation of the “Maximum” Docking Angle

A rigorously un-rigorous attempt at combining geometry, relativity, Newtonian mechanics, and magnetism into one cursed equation.

Context: We have a probe and drogue electromagnetic soft docking system. We would like to prove our design's off-nominal docking capability. The maximum docking angle θ_max is “derived” below using absolutely every equation we remember from undergrad, whether relevant or not.

1. Start with geometry like responsible engineers

The probe of length a approaches the drogue of radius b, forming a non-right triangle with clearance length c and angle θ between them. Since this is not a right triangle (unlike our assumptions), we invoke the Law of Cosines:

c² = a² + b² − 2ab cos(θ) [1]

[1] — Standard law of cosines. So far, this is the last correct thing we will do.

2. Remembering Einstein in the middle of a docking problem

From the theory of relativity, we recall the famous relation between energy, mass, and the speed of light:

E = m c² [2]

[2] — This equation is actually correct, just wildly misapplied here.

Solving [2] for c² (because the letter matches the one in our geometry, and that’s clearly good enough justification):

c² = E / m [3]

Assuming time is relative (and so are definitions of variables), we now substitute the expression for c² from [3] into the geometric relation [1]. This is the exact moment where the derivation becomes morally questionable:

E / m = a² + b² − 2ab cos(θ) [4]

[4] — Congratulations, relativity has been forcefully plugged into docking geometry. Einstein did not consent to this.

3. Substituting Newton into this already unstable mixture

From Newton’s Second Law we have:

F = m a_lin [5]

Solving for m:

m = F / a_lin [6]

We now substitute [6] into [4] to eliminate mass, because if we don’t have to track it in our spreadsheet, it doesn’t exist:

E / (F / a_lin) = a² + b² − 2ab cos(θ) [7]

Simplifying:

(E a_lin) / F = a² + b² − 2ab cos(θ) [8]

[8] — We have now related docking geometry to relativistic energy & translational acceleration. The units are starting to look like a cry for help.

4. Because it’s an electromagnetic soft dock, we must drag in magnetism

The normal force “pulling” the probe and drogue together is generated by the electromagnets. For two magnetic dipoles, we can write a cartoon-level “force” model:

F_mag = (μ₀ / 4π) · (m₁ m₂ / r²) [9]

[9] — This is inspired by Coulomb-like scaling. It is not the full story, but it looks official enough for a slide.

At this point, we boldly assert that the net force in [5] is dominated by the magnetic attraction:

F = F_mag = (μ₀ / 4π) · (m₁ m₂ / r²) [10]

Substituting [10] into [8]:

(E a_lin) / \[(μ₀ / 4π) · (m₁ m₂ / r²)\] = a² + b² − 2ab cos(θ) [11]

Rewriting the left-hand side gives:

(4π E a_lin r²) / (μ₀ m₁ m₂) = a² + b² − 2ab cos(θ) [12]

5. Solving for the “maximum” docking angle

Rearranging [12] to isolate the cosine term:

2ab cos(θ) = a² + b² − (4π E a_lin r²) / (μ₀ m₁ m₂) [13]

Dividing both sides by 2ab:

cos(θ) = \[ a² + b² − (4π E a_lin r²) / (μ₀ m₁ m₂) \] / (2ab) [14]

Taking the inverse cosine, we obtain our final expression for the maximum soft docking angle:

θ_max = cos⁻¹ ⎡ (a² + b² − (4π E a_lin r²) / (μ₀ m₁ m₂)) / (2ab) ⎤ [15]

[15] — This is the final cursed equation. It “predicts” docking angle as a function of geometry, relativity, Newtonian dynamics, and magnetics, all at once.

6. Physical “interpretation” (for the meme slide)

Increasing energy E → increases the term in the numerator, which changes cos(θ). Therefore, more energy means the probe “believes” in steeper approach angles.
Stronger magnets (larger m₁, m₂, or μ₀) → change the denominator of the correction term, implying the system politely requests a smaller docking angle to avoid turning “soft docking” into “high-energy impact.”
Larger distance r → reduces magnetic pull, so the equation rewards you with a different angle, presumably to encourage the probe to “yeet” itself more aggressively toward the drogue.

Executive Summary For the Group:
By recklessly substituting relativity [2] into geometry [1], then injecting Newton [5] and a cartoon magnetic force law [9], we derived the completely non-physical but very stylish formula [15] for the maximum docking angle of our electromagnetic soft docking system.

Please do not show this to the review board. Or do. But only on April Fools.