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Problem

The monster $ M$ wants to catch the elusive tofubeast $ T$.

$ M$ and $ T$ are two points on a plane. $ T$ is moving in direction $ a_{T}$ (an angle clockwise relative to $ 0^{\circ}$ north) at a constant velocity $ v_{T}$. $ a_{S}$ is the direction from $ M$ to $ T$ (angle of sight, relative to the same $ 0^{\circ}$ north), $ d$ is the distance between them, and $ v_{M}$ is the constant velocity of point $ M$. All these are given, and shown in Figure 1. Angles $ \theta $ and $ \alpha $ are derived (from givens, and from the unknown direction $ a_{M}$ in which $ M$ will travel):

$\displaystyle \theta$ $\displaystyle = a_{T} - (a_{S} - 180)$    
$\displaystyle \alpha$ $\displaystyle = a_{S} - a_{M}$    

Figure 1: The physical problem.
Image problem-physical

Suppose a point $ I$ such that if $ T$ and $ M$ continue at their constant speeds and directions they will intercept at time $ t_{0}$. In other words, at any point in time $ t$, with $ \Delta t = t_{0} - t$ seconds until interception, the points can be represented as shown in Figure 2, where $ r_{XY}$ is the rate at which points $ X$ and $ Y$ are approaching. (Notice in particular the way assumptions of the problem are captured: an extant $ I$, and $ \theta $ and $ \alpha $ constant over time.)

Figure 2: A mathematical representation of the problem.
Image problem-representational

The problem:

  1. Make an expression for $ \alpha $ in terms of $ v_{T}$, $ v_{M}$, and $ \theta $.

  2. How many points $ I$ are there for a given $ v_{T}$, $ v_{M}$, and $ \theta $?


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