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Linear Motion
------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0032.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Linear Motion
A car traveling on a straight line with a fixed rate c, = over the time period [0, t],
------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0036.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Linear Motion
A car traveling on a straight line with a fixed rate c, over the time period [0, t], covers the distance = ct.
------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0033.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Linear Motion
A car traveling on a straight line with a fixed rate c, over the time period [0, t], covers the distance = ct.
If c is positive,
c is positive
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Linear Motion
A car traveling on a straight line with a fixed rate c, over the time period [0, t], covers the distance = ct.
If c is positive, ct is positive, so the car moves to = the right by a distance of ct.
c is positive
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Linear Motion
A car traveling on a straight line with a fixed rate c, over the time period [0, t], covers the distance = ct.
If c is positive, ct is positive, so the car moves to = the right by a distance of ct.
c is positive
Note that ct =3D        = ;      is also the signed area under the rate function r(t) =3D c from 0 to t.
∫
a
b
f(x) dx
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Linear Motion
A car traveling on a straight line with a fixed rate c, over the time period [0, t], covers the distance = ct.
rate=
time=
ct= > 0
c
If c is positive, ct is positive, so the car moves to = the right by a distance of ct.
c is positive
Note that ct =3D        = ;      is also the signed area under the rate function r(t) =3D c from 0 to t.
r(t) =3D c&= gt;0
area= =3D ct >0 is the distance travel to the right
t
0
∫
a
b
f(x) dx
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If c = is negative then the car moves to the left by |ct|.
Linear Motion
c is negative
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If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
c is negative
Linear Motion
∫
a
b
f(x) dx
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c is negative
rate=
time=
ct= < 0
c
r(t) =3D c= <o
t
0
Linear Motion
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
∫
a
b
f(x) dx
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c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
b
f(x) dx.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
r(t) =3D c= <o
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c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
r(t) =3D c= <o
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Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
r(t) =3D c= <o
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Divid= e [a, b] into a regular partition with n subintervals
Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
r(t) =3D c= <o
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Divid= e [a, b] into a regular partition with n subintervals
Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
=
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
a
b
t i-1
t i
t1
t2
t n–1
r(t) =3D c= <o
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file:///C:/E12BB24E/101LinearMotion_files/slide0057_image063.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AAoAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0052.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
=
*
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
a
b
t i-1
t i
t1
t2
Divide [a, b] into a regular partition with n subintervals <= span style=3D'font-size:156%'>and let ti be a point in the subinterval [ti= -1, ti]
t n–1
r(t) =3D c= <o
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file:///C:/E12BB24E/101LinearMotion_files/slide0052_image078.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AAoAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0054.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
=
*
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
a
b
t i-1
t i
t1
t2
Divide [a, b] into a regular partition with n subintervals <= span style=3D'font-size:156%'>and let ti be a point in the subinterval [ti= -1, ti]
t1
*
t n–1
r(t) =3D c= <o
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file:///C:/E12BB24E/101LinearMotion_files/slide0054_image093.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhAwALAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAQAB AAoAgAAAAAAAAAIDjI8FADs= ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0054_image094.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBwAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAF AAUAgQAAAAAAAP8AAAECAwIIRG6ip8JrUAEAOw== ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0055.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
=
*
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
a
b
t i-1
t i
t1
t2
Divide [a, b] into a regular partition with n subintervals <= span style=3D'font-size:156%'>and let ti be a point in the subinterval [ti= -1, ti]
t1
t2
*
*
t n–1
r(t) =3D c= <o
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charset="us-ascii" Slide 1
Now suppose the rate r(t) changes continuously with respect to time over the interval of time a < = t < b.  
=
*
c is negative
Hence t= he signed distance traveled is&nbs= p;
∫
a
b
f(x) dx.
The positive answer means it’s displaced to the right and negative answer means it’s displaced to= the left.
Linear Motion
rate=
time=
ct= < 0
c
t
0
∫
a
b
f(x) dx
If c is negative then the car moves to the left by |ct|.
Note that the signed area ct =3D        = ;    , between
r(t) and the x-axis, is negative.
a
b
t i-1
t i
t1
t2
Divide [a, b] into a regular partition with n subintervals <= span style=3D'font-size:156%'>and let ti be a point in the subinterval [ti= -1, ti].
ti*
t1
t2
*
*
t n–1
r(t) =3D c= <o
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R0lGODlhBwAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAG AAUAgQAAAAAAAP8AAAECAwIKRB5gIhnMXFQoFQA7 ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0056_image130.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBwAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAF AAUAgQAAAAAAAP8AAAECAwIIRG6ip8JrUAEAOw== ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0004.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
a
b
t i-1
t i
t1
t2
r =3D r(t)
Δt
Linear Motion
Δt =3D
b R= 11; a
n
t n–1
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a
b
t i-1
t i
t2
*
t1
t2
ti
*
r =3D r(t)
t1
*
Δt
r(ti)
*
Linear Motion
Δt =3D
b R= 11; a
n
t n–1
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Content-Type: image/gif R0lGODlhJgACAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAEAAAAk AAEAgAAAAAAAAAIFjI+pywUAOw== ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0053_image161.gif Content-Transfer-Encoding: base64 Content-Type: image/gif R0lGODlhBwAHAHcAMSH+GlNvZnR3YXJlOiBNaWNyb3NvZnQgT2ZmaWNlACH5BAEAAAAALAAAAAAF AAUAgQAAAAAAAP8AAAECAwIIRG6ip8JrUAEAOw== ------=_NextPart_01C97CE7.A25B1EA0 Content-Location: file:///C:/E12BB24E/101LinearMotion_files/slide0005.htm Content-Transfer-Encoding: quoted-printable Content-Type: text/html; charset="us-ascii" Slide 1
a
b
t i-1
t i
t2
*
t1
t2
ti
*
r =3D r(t)
t1
*
Then r(t= i= ) = 16;t is the approximate distance the car = traveled during the time interval [ti= -1, ti] .<= /span>
*=
Δt
r(ti)
*
Linear Motion
Δt =3D
b R= 11; a
n
t n–1
r(ti)Δt
*
*=
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a
b
t i-1
t i
t2
*
t1
t2
ti
*
r =3D r(t)
t1
*
i=3D= 1
n=
W= 21; r(ti)Δt