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Up: Two-Dimensional Interception Problem Previous: Sensible Solutions for the

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.


    $\displaystyle a_{T}$ $\displaystyle = 90^{\circ}$    [east]    
    $\displaystyle a_{S}$ $\displaystyle = 225^{\circ}$    [southwest]    
    $\displaystyle v_{T}$ $\displaystyle = v_{M} = 10$    mph    


    Image example-acute10

    Solution

    $\displaystyle \theta$ $\displaystyle = a_{T} - \left( a_{S} - 180 \right) = 45^{\circ}$    
    $\displaystyle \alpha$ $\displaystyle = \arcsin \left( \sin \theta \cdot \frac{ v_{T} }{ v_{M} } \right) = 45^{\circ}$    
    $\displaystyle a_{M}$ $\displaystyle = a_{S} - \alpha = 180^{\circ}$   [south]    

    Variations

  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

    $\displaystyle \theta = 135^{\circ}$    
    $\displaystyle \alpha = 40^{\circ}$    
    $\displaystyle a_{M} = 185.0^{\circ}$    


    Image example-obtuse

    Variations

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

    $\displaystyle \alpha$ $\displaystyle = \arcsin \left( \sin \theta \cdot \frac{ v_{T} }{ v_{M} } \right) = \arcsin \left( 0.707 \cdot \frac{10}{9} \right) = 51.8^{\circ}$    
    $\displaystyle a_{M}$ $\displaystyle = 225^{\circ} - \alpha = 173.2^{\circ}?$    

    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

    $\displaystyle \theta$ $\displaystyle = 270^{\circ}$    or $\displaystyle -90^{\circ}$    
    $\displaystyle \alpha$ $\displaystyle = -30^{\circ}$    
    $\displaystyle a_{M}$ $\displaystyle = 255.0^{\circ}$    


    Image example-negativeside

    Variation

  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

    $\displaystyle \theta$ $\displaystyle = 225^{\circ}$    or $\displaystyle -135^{\circ}$    
    $\displaystyle \alpha$ $\displaystyle = -28.1^{\circ}$    
    $\displaystyle a_{M}$ $\displaystyle = 343.1^{\circ}$    


    Image example-lineofsight

    Variations


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