The More, The Mightier (Part 2)

Calculus Level 4

Lanchester's Square Law can be used to roughly describe the way in which two opposing military forces change over time during battle. Suppose the number of troops in "Force A" is $A$ , and the number of troops in "Force B" is $B$ .

The rates of change in troop strength (numbers of troops) over time are given by:

$\dfrac{dA}{dt} = -\beta B \hspace{1cm}$ and $\hspace{1cm} \dfrac{dB}{dt} = -\alpha A$ .

Constants $\alpha$ and $\beta$ represent the relative fighting proficiencies of "Force A" and "Force B", respectively.

Suppose that $A=1$ and $B=2$ at time $(t=0)$ . Suppose also that $(\beta = 1)$ .

Determine the value of $\alpha$ such that the two sides fight each other for eternity, with neither side's troop strength ever being entirely reduced to zero.

Details and Assumptions: Assume that $A$ and $B$ can vary continuously.

The answer is 4.

1 solution

Steven Chase
Oct 6, 2016

The analytical derivation is shown below. It is also quite trivial to check this numerically with a computer. When $\alpha = 3.99$ , "Force B" just barely wins out over "Force A". When $\alpha = 4.01$ , the reverse is true. When $\alpha = 4.0$ , the program runs forever (and eats up all the RAM in your computer if you are programming it in Python).

Python Code

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Talulah Riley - 8 months, 2 weeks ago

The More, The Mightier (Part 2)

The answer is 4.

1 solution

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