alittlebitofcalculus-pdf-january2011-111112001007-phpapp01

# 543210987654321 2109876543210987654321

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Unformatted text preview: . 182 We need to be more clear about the terms CHANGE in position, displacement , displacement and distance travelled in A) 1-Dimension in fixed direction , B) fixed axis distance fixed fixed or 1-Dimension with CHANGE in direction and C) 2-Dimensions. These terms need to be explained from both the Calculus point of view and the Physics (what is happening in nature) point of view and how they are related. A) 1-Dimension in fixed direction : in the example the car travelled East in a fixed fixed direction, that is to say moving along a straight line without reversing or changing direction. So in this very speical case from the Physics point of view: Physics CHANGE in position = displacement = distance travelled . displacement distance Also, from the Calculus point of view, over a given interval, say [t 1 , t 2 ] : CHANGE in position = t2 ∫ t1 x’(t)dt = area under the speed cur ve area curv ar This is because the SPEED function did NOT CHANGE SIGN. In 1-Dimensional motion the SPEED function will CHANGE SIGN only if there is a CHANGE in direction. B) fixed axis or 1-Dimension with CHANGE in direction : Let us review an example fixed of the projected ball. 1-Dimension y(t1) y(T/ 2) y’(t1) y(t2) 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 0987654321 1 0987654321 098765432654321 654321 654321 654321 654321 654321 654321 T/ 2 (0,0) t2 t1 y’(t2) T t vertical speed y ’(t) = u . s i n θ -- g t 183 CHANGE in position over [t1, t2] = dispalcement = y (t2) -- y(t1) distance travelled =| ∫ T /2 t1 t2 ∫ t1 y ’(t)dt = y(t2) -- y(t1) y ’(t)dt| + | ∫ t2 T/ 2 y ’(t)dt| = |y(T/ 2 )| -- y(t1) | + | y(t2) -- y(T/ 2)| The distance travelled (odometer reading or length of path) in the usual (nondistance mathematical) sense is used to convey the notion “CHANGE in position” But position”. in Calculus these two are different. “CHANGE in position” is always with reference to a co-ordinate system. If an object moves in a circle o...
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