| The Monkey And The Hunter |
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The essentials of the problem are stated in many introductory guides to Physics , such as Caltech 's '' The Mechanical Universe '' television series or Gonick and Huffman's ''Cartoon Guide to Physics.'' In essence, the problem is as follows: a hunter with a blowgun goes out in the woods to hunt for monkeys and sees one hanging in a tree, at the same level as the hunter's head. The monkey, we suppose, releases its grip the instant the hunter fires his blowgun. Where should the hunter aim and when should he fire in order to hit the monkey? To answer this question, recall that according to , not upon its velocity in the horizontal direction. (This can easily be treated by representing velocity and acceleration as Vector s in a Cartesian Coordinate System .) The hunter's dart, therefore, falls with the same acceleration as the monkey. Assume for the moment that gravity was not at work. In that case, the dart would proceed in a straight-line trajectory at a constant speed ( Newton's First Law ). Gravity causes the dart to fall away from this straight-line path, making a trajectory which is in fact a Parabola . Now, consider what happens if the hunter aims directly at the monkey, and the monkey releases his grip the instant the hunter fires. Because the force of gravity accelerates the dart and the monkey equally, they fall the same distance in the same time: the monkey falls from the tree branch, and the dart falls ''the same distance'' from the straight-line path it would have taken in the absence of gravity. Therefore, ''the dart will always hit the monkey,'' no matter the initial speed of the dart. To write equations for the motion of the monkey and the hunter's dart, use ''g'' to denote the acceleration of gravity, ''t'' for elapsed time and ''h'' for the initial height of the monkey. Using ''VY0'' to denote the initial vertical speed of the dart, the equations for the vertical motion (altitude) of the dart and the monkey are respectively : and : They will collide when those altitudes are the same, that is : The term ''gt&2 /2'' is both present on both sides of the equation, which then can be simplified to : Given a non-zero it can be rewritten to define when that occurs: : And given a zero the only possible values that satisfies the equation are ''h'' = 0 and any value of ''t.'' In short, there is always a time ''t'' when both the dart and the monkey will collide vertically. EXTERNAL LINKS
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