alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

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Unformatted text preview: f radius r thru one complete circle, then the CHANGE in position = 0. But the distance travelled = 2πr . distance C) 2-Dimensions: What if the object is moving in 2-Dimension as in the case of a projected ball ? Height y(t) 2-Dimension (0, 0) P2 P1 Range x(t) This type of problem may be split into two 1-Dimension problems. x’(t) = u . cos θ 1-Dimension in fixed direction. 1-Dimension fixed y ’(t) = u . s i n θ -- g t 1-Dimension with CHANGE in direction. 1-Dimension CHANGE in position is from P1 = (x(t1), y(t1)) to P2 = (x(t2) , y(t2)) = P2 − P1 ½ displacement = {[x(t2) − x(t1)]2 + [y(t2) − y(t1)]2 } 184 1 y(t) = u.sin θ. t -- -- g t 2 . 2 3: Example 3 What is the distance travelled (length of path) in 2-Dimension ? distance dx = infinitesimal CHANGE in range. dy = infinitesimal CHANGE in height. ds = infinitesimal piece of the path. ds dx dy (0, 0) x(t) = u.cos θ. t . dx ----- = x’(t) = u.cos θ . So dx = x’(t)dt = u.cos θ. dt . . dt dy ----- = y’(t) = u . sin θ -- g t . So dy = y’(t)dt = (u . sin θ -- g t) . dt dt ½ ½ By Pythagoras theorem : ds = [(dx) 2 + ( dy)2 ] = [(x’(t))2 + (y’(t))2] . dt ½ s = ∫ d s = ∫ [ (x’(t)) 2 + (y’(t)) 2] . dt We should now look up the Table of Integr als to find the appropriate integral Integ where the integrand from the given x(t) and y(t) is in the above form. In the case of the vertical dimension of the projected ball the range x(t) is absent. So the above formula reduces to : s = ∫ d s = ∫ y’(t). dt . Adapting this technique to find the distance travelled ( length of flight path) of a distance plane gliding forward at the rate of 500 feet/ second and downward at the rate of 20 feet/ second (see page 82) we note that : dx ----- = x’(t) = 500 feet/ second. So dx = 500 . dt dt dy ----- = y’(t) = −20 feet/ second. So dy = −20 . dt dt ½ ½ s = ∫ d s = ∫ [(x’(t)) 2 + ( y’(t)) 2 ] . dt = ∫ [ (500) 2 + (−20) 2 ] . dt 185 Let us now apply this method to find the length of...
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This note was uploaded on 11/29/2012 for the course PHYSICS 105 taught by Professor Tamerdoğan during the Fall '09 term at Middle East Technical University.

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