Create function:
f(m)=v-(gm/c)(1-e⁻⁽ᶜʼᵐ⁾ᵗ), and find its derivative f'(m):
-(g/c)(1-e⁻⁽ᶜʼᵐ⁾ᵗ)-(gm/c)(-(ct/m²)e⁻⁽ᶜʼᵐ⁾ᵗ),
-(g/c)[1-e⁻⁽ᶜʼᵐ⁾ᵗ-(ct/m)e⁻⁽ᶜʼᵐ⁾ᵗ],
-(g/c)[1-e⁻⁽ᶜʼᵐ⁾ᵗ(1+ct/m)].
Apply Newton’s Method:
mᵣ₊₁=mᵣ-f(mᵣ)/f'(mᵣ).
We need a starting value m₀.
A realistic value might be 100kg, so m₀=100.
Substituting m₀=100 (r=0) for mᵣand the expressions for f and f', we can find successive iterations for m.
m₁=98.27610168.
Now plug in mᵣ=m₁:
m₂=98.29327874, and so on:
m₃=98.29428047,
m₄=98.29428047, so we have found a fairly accurate solution, m=98.29428 (error less than 0.00001).
To check the solution, we should find that
v=52.75645=(gm/c)(1-e⁻⁽ᶜʼᵐ⁾ᵗ) when m=98.29428kg.
Reducing this by 5kg for the weapon and we get the appropriate mass of the paratrooper to be 93.29428kg.