2005 06 15 lecture notes graphical motion-1

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RECTILINEAR KINEMATICS: GRAPHICAL

PIECEWISE CONTINUOUS (OR ERRATIC) MOTIONObjectives:

Students will be able to

determine position, velocity,

and acceleration of a particle

using graphs.

Content:

• Applications

• s-t, v-t, a-t, v-s, and a-s

diagrams

• Concept quiz

• roup problem solving

• Attention quiz

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APPLICATION

!n many e"periments, a

velocity versus position #v-

s$ profile is obtained.

!f we have a v-s graph for

the roc%et sled, can we

determine its acceleration at

 position s & '(( meters )

*ow)

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GRAPHING

raphing provides a good way to handle comple"

motions that would be difficult to describe with

formulas. raphs also provide a visual description of

motion and reinforce the calculus concepts of

differentiation and integration as used in dynamics.

+he approach builds on the facts that slope and

differentiation are lin%ed and that integration can be

thought of as finding the area under a curve.

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ST GRAPH

lots of position vs. time can beused to find velocity vs. time

curves. inding the slope of the

line tangent to the motion curve at

any point is the velocity at that

 point #or v & dsdt$.

+herefore, the v-t graph can be

constructed by finding the slope atvarious points along the s-t graph.

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!T GRAPH

lots of velocity vs. time can be used

to find acceleration vs. time curves.inding the slope of the line tangent to

the velocity curve at any point is the

acceleration at that point #or a & dvdt$.

+herefore, the a-t graph can be

constructed by finding the slope at

various points along the v-t graph.

Also, the distance moved

#displacement$ of the particle is the

area under the v-t graph during time ∆t.

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AT GRAPH

iven the a-t curve, the change

in velocity #∆v$ during a time

 period is the area under the a-t

curve.

So we can construct a v-t graph

from an a-t graph if we %now the

initial velocity of the particle.

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AS GRAPH

+his equation can be solved for v/, allowing you to solve for

the velocity at a point. 0y doing this repeatedly, you can

create a plot of velocity versus distance.

A more comple" case is presented bythe a-s graph. +he area under the

acceleration versus position curve

represents the change in velocity

#recall ∫  a ds & ∫  v dv $.

a-s graph

1 #v/2 3 vo2$ & & area under the∫ s4

s/

a ds

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!S GRAPH

Another comple" case is presented

 by the v-s graph. 0y reading the

velocity v at a point on the curve

and multiplying it by the slope of

the curve #dvds$ at this same point,we can obtain the acceleration at

that point.

a & v #dvds$

+hus, we can obtain a plot of a vs. s

from the v-s curve.

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E"AMPLE

Given:  v-t graph for a train moving between two stations

#in$:  a-t graph and s-t graph over this time interval

+hin% about your plan of attac% for the problem5

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E"AMPLE (contin%e$)

So&%tion: or the first '( seconds the slope is constant

and is equal to6  a(-'(  & dvdt & 7('( & 7' fts4

7

-7'

'

a#fts4$

t#s$

Similarly, a'(-8( & ( and a8(-/4( & -7' fts4

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E"AMPLE (contin%e$)

+he area under the v-t graph

represents displacement.

∆s(-'( & 1 #7($#'($ & 9(( ft

∆s'(-8( & #9($#7($ & 47(( ft

∆s8(-/4( & 1 #7($#'($ & 9(( ft9((

'(((

'9((

'( 8( /4(t#s$

s#ft$

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CONCEPT 'UI

/. !f a particle starts from rest and

accelerates according to the graph

shown, the particle:s velocity at

t & 4( s is

A$ 4(( ms 0$ /(( ms

C$ ( ;$ 4( ms

4. +he particle in roblem / stops moving at t & <<<<<<<.

A$ /( s 0$ 4( s

C$ '( s ;$ 7( s

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GROUP PROLEM SOL!ING

Given:  +he v-t graph shown

#in$: +he a-t graph, average

speed, and distance

traveled for the '( s

interval

P&*n: ind slopes of the curves and draw the v-t graph. ind

the area under the curve--that is the distance traveled.

inally, calculate average speed #using basic

definitions5$.

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GROUP PROLEM SOL!ING

So&%tion:

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GROUP PROLEM SOL!ING (contin%e$)

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ATTENTION 'UI

/. !f a car has the velocity curve shown, determine the time t

necessary for the car to travel /(( meters.

A$ = s 0$ 7 s

C$ /( s ;$ 9 s

t

v

+ s

,-

t

v

4. Select the correct a-t graph for the velocity curve shown.

A$ 0$

C$ ;$

a

t

a

t

a

t

a

t

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