Unformatted text preview: .
182 We need to be more clear about the terms CHANGE in position, displacement ,
and distance travelled in A) 1-Dimension in fixed direction , B) fixed axis
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 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:
CHANGE in position = displacement = distance travelled .
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
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
of the projected ball.
y(t1) y(T/ 2) y’(t1) y(t2) 0987654321
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
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...
View Full Document
- Fall '09
- Limit, Δx