My Math Forum body moving on acoordinate line with s in meters and t in se

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 November 5th, 2010, 07:48 AM #1 Senior Member   Joined: Oct 2009 Posts: 895 Thanks: 1 body moving on acoordinate line with s in meters and t in se Posision s = (t^4/4) - t^3 + t^2 , 0<=t<=3 this body moving on acoordinate line with s in meters and t in second . when if ever during the intreval dose the body change direction ? How I solve this quetion .... I do derivatives but after that what i do ?
 November 5th, 2010, 08:33 AM #2 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,963 Thanks: 1148 Math Focus: Elementary mathematics and beyond Re: body moving on acoordinate line with s in meters and t i $s\,=\,\frac{t^4}{4}\,-\,t^3\,+\,t^2,\,0\,\le\,t\,\le\,3$ $\frac{ds}{dt}\,=\,t^3\,-\,3t^2\,+\,2t$ (velocity). $\frac{ds}{dt}\,=\,t(t^2\,-\,3t\,+\,2)$ $\frac{ds}{dt}\,=t(t\,-\,1)(t\,-\,2)$ (0, 1) - positive (1, 2) - negative (2, 3] - positive The body changes direction when velocity goes from negative to positive or vice-versa, so the body changes direction at 1 second and at 2 seconds.
November 5th, 2010, 10:44 AM   #3
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Re: body moving on acoordinate line with s in meters and t i

Hello, r-soy!

greg1313 did all the work for us.
Let me give you the background explanation . . .

Quote:
 $\text{Position: }\;s \:=\:\frac{1}{4}t^4\,-\,t^3 \,+\, t^2,\;\;0\,\le\,t\,\le\,3$ $\text{This body moving on a coordinate line with }s\text{ in meters and }t\text{ in seconds.}$ $\text{When, if ever, during the interval does the body change direction ?}$

In order for a body to change direction, it must first come to a full stop,
[color=beige]. . [/color]Hence, we are seeking times when the velocity is zero.

$\text{The velocity is: {\:v \:=\:s#39; \:=\:t^3\,-\,3t^2\,+\,2t$

$\text{Then: }\:t^3\,-\,3t^2\,-\,2t \:=\:0 \;\;\;\Rightarrow\;\;\;t(t\,-\,1)(t\,-\,2) \:=\:0 \;\;\;\Rightarrow\;\;\;t \:=\:0,\,1,\,2$

$\text{Therefore, the body changes directions at: }\,t \:=\:0,\,1,\,2\text{ seconds.}$

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