Examples

1. Acute approach. The Tofubeast is running due east at 10 mph. Monster is northeast of the Tofubeast and can run at 10 mph.

   
   

   $$a_{T} = 90^{\circ}$$

   $$a_{S} = 225^{\circ}$$

   $$v_{T} = v_{M} = 10$$

   
   

   
   

   

   Solution

   $$\theta = a_{T} - \left( a_{S} - 180 \right) = 45^{\circ}$$

   $$\alpha = \arcsin \left( \sin \theta \cdot \frac{ v_{T} }{ v_{M} } \right) = 45^{\circ}$$

   $$a_{M} = a_{S} - \alpha = 180^{\circ}$$

   
   

   Variations

   • if $v_{T} = 5$ then $a_{M} = 204.3^{\circ}$ [south-southwest]
   • if $v_{T} = 14$ then $a_{M} = 143.1^{\circ}$ [southeast]
     
   • if $v_{T} = 15$ then $a_{M} =$ undefined
     (notice if $\alpha = 90^{\circ}$, then $10 \cdot \sqrt{2} = 14.14 < 15$)

2. Obtuse approach. The Tofubeast is running due south at 10 mph. Monster is northeast of the Tofubeast and can run at 11 mph.

   
   Solution

   $$\theta = 135^{\circ}$$

   $$\alpha = 40^{\circ}$$

   $$a_{M} = 185.0^{\circ}$$

   
   

   
   

   

   Variations

   • if $v_{M} = 15$ then $a_{M} = 196.9^{\circ}$
   • if $v_{M} = 20$ then $a_{M} = 204.3^{\circ}$
   • if $v_{M} = 30$ then $a_{M} = 211.4^{\circ}$
   • if $v_{M} = 9$ then $a_{M} =$ undefined

   Notice in the last variation, where $v_{M} = 9$, then according to equation (1):

   $$\alpha = \arcsin \left( \sin \theta \cdot \frac{ v_{T} }{ v_{M} } \right) = \arcsin \left( 0.707 \cdot \frac{10}{9} \right) = 51.8^{\circ}$$

   $$a_{M} = 225^{\circ} - \alpha = 173.2^{\circ}?$$

   
   
   This would be a poor choice of $\alpha$: $M$ is instructed to head away from $T$ in order to intercept, which makes no sense. But $\theta > 90^{\circ}$ and $v_{T} > v_{M}$, which is the special case of non-interception even when equation (1) is defined, and so $a_{M}$ is undefined.

3. Negative-side approach. The Tofubeast is running northwest at 5 mph. Monster is northeast of the Tofubeast and can run at 10 mph.

   
   Solution

   $$\theta = 270^{\circ} -90^{\circ}$$

   $$\alpha = -30^{\circ}$$

   $$a_{M} = 255.0^{\circ}$$

   
   

   
   

   

   Variation

   • if $a_{T} = 0^{\circ}$, $v_{T} = 10$ then $\theta = -45^{\circ}$, $a_{M} = 270^{\circ}$

4. Angle of sight. The Tofubeast is running north at 10 mph. Monster is southeast of the Tofubeast and can run at 15 mph.

   
   Solution

   $$\theta = 225^{\circ} -135^{\circ}$$

   $$\alpha = -28.1^{\circ}$$

   $$a_{M} = 343.1^{\circ}$$

   
   

   
   

   

   Variations

   • if $a_{S} = 345^{\circ}$ then $\theta = -165^{\circ}$, $a_{M} = 354.9^{\circ}$
   • if $a_{S} = 0^{\circ}$ then $\theta = 180^{\circ}$, $a_{M} = 0^{\circ}$
   • if $a_{S} = 45^{\circ}$ then $\theta = 135^{\circ}$, $a_{M} = 16.9^{\circ}$
   • if $a_{S} = 90^{\circ}$ then $\theta = 90^{\circ}$, $a_{M} = 48.2^{\circ}$
   • if $a_{S} = 115^{\circ}$ then $\theta = 65^{\circ}$, $a_{M} = 77.8^{\circ}$