kinematics 1d part1 - page not found | university of alberta · one dimensional kinematics –...
TRANSCRIPT
One Dimensional Kinematics – Motion Along a Line!
Concepts at a Glance!
- distance!
- displacement!
- velocity and speed!
- acceleration!
- special case: constant acceleration ! (incl. free fall)!
- position and velocity from integration (i.e. anti-differentiation)!
!
"! x =! x f #! x 0
!
! v = lim
"t#0
"! x
"t=
d! x
dt
!
! a = lim
"t#0
"! v
"t=
d! v
dt
!
! v
x=! v 0x
+! a
xdt
0
t
"
!
! x =" x 0
+! v
xdt
0
t
"
!"#$%&#'speed=distance/(elapsed time)'''''''()*'+%&,-./0#'12'%"#$%&#'velocity!
Instantaneous velocity!
Instantaneous acceleration!
3'
4-567%8#+#,.'
!!'91:1,'-+67-#5'%'8;%,&#'-,'615-:1,<'.;-5'8;%,&#'-5'
=,1>,'%5'.;#'0-567%8#+#,.'
!! 4-567%8#+#,.'-5'%'"#8.1$'.;%.'61-,.5'2$1+'%,'1?@#8.A5'-,-:%7'
615-:1,'.1'-.5'B,%7'615-:1,'
!!(1.%:1,'
!! C%$&#'8;%,'-,'%'D/%,:.E'%$#'&#,#$%77E'0#,1.#0'/5-,&'.;#'
8%6-.%7'&$##='7#F#$'4#7.%G'H'
!! I15-:1,'-5'1J#,'0#,1.#0'%5'AKA'51'0-567%8#+#,.'8%,'?#'>$-F#,'
AH!A'
!! L'>-77'/5#'AH!A'1$'A"A'21$'0-567%8#+#,.'%,0'A!A'21$'615-:1,'
!! (1.#<'.;#'?11='/5#5'?1.;'H!'%,0#!M!N'21$'0-567%8#+#,.5'>;#$#'!
N'-5'.;#'
-,-:%7'615-:1,'
O'
P#718-.E'
!!P#718-.EG'7-=#'0-567%8#+#,.G'-5'%'"#8.1$'
!! -Q#Q'L.'0#58$-?#5'+1:1,'-,'%'&-"#,'0-$#8:1,'
!! R6##0'S!"#$%"$%"&'(#T'-5'.;#'+%&,-./0#'12'"#718-.E'%,0'-5'%'
58%7%$'
!!U1,5-0#$'%'$/,,#$'5.%$:,&'%.'%'615-:1,'KN'%.':+#'.
NG'
#,0-,&'%.':+#'.'%,0'615-:1,'K'
!!*;#'%"#$%&#'"#718-.E'12'.;#'$/,,#$'-5'&-"#,'?E'
)*+
$*+
$+
)+
!
average velocity : v =displacement
time elapsed="! x
"t=
! x #! x 0
t # t0
R-&,'12'.;#'%"#$%&#'"#718-.E'-5'$#7%:"#'.1'E1/$'$#2#$#,8#'2$%+#Q''R.%.#'.;#'%55/+6:1,V''W'
R6##0'
Start Finish
I can run 1 minute and 20 sec for a 400m sprint. What is my average speed for this exercise?!
!Q!'N'+X5'
YQ!'NQON'+X5'
UQ!'MNQZN'5X+'
4Q!'Z'+X5'
[Q!'MZ'+X5'
start!
\'
P#718-.E'
!!'
!Q!'N'+X5'
YQ!'NQON'+X5'
UQ!'MNQZN'5X+'
4Q!'Z'+X5'
[Q!'Z'5X+'
I can run 1 minute and 20 sec for a 400m sprint. What is my average velocity for this exercise?!
start!
Z'
T-X plots (Velocity)!
!!]$%0-#,.'12'%'5.$%-&;.'7-,#'-5<'
!!']$%0-#,.'-5'@/5.<'
*;#'5716#'S1$'&$%0-#,.T'
1,'%'615-:1,'"5Q':+#''
671.'-5'@/5.'.;#'%"#$%&#'
"#718-.EV'
!
Gradient ="y
"x
!
Gradient ="x
"t= velocity
^'
Problem: Interpreting Plots!
!!_;-8;'12'.;#5#'671.5'$#6$#5#,.5'%'81,5.%,.G',#&%:"#'"#718-.E`'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
B A
C
D
a'
Problem: Interpreting Plots!!!Which of these plots represents an object which moves with a constant velocity out, stops for a short while and then returns with a constant velocity?!
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
I15-:1,G'K'
B
A
C
D
*-+#G'.'b'
Instantaneous Velocity!!! Velocity varies as a function of time, hence it is important to describe the velocity at a given instant of time.!
!!'The idea starts with the same formula:'
!
Average velocity v ="! x
"t
!! Now shrink the time interval to zero! (take limit):!
!
Instantaneous velocity ! v = lim
"t#0
"! x
"t=
d! x
dt
c'*-+#G'.'
I15-:1,G'K'
A QQQ'
!
"x
!
"t
*-+#G'.'
I15-:1,G'K'
A
!
"x
!
"t
I$1?7#+<'L,.#$6$#:,&'I71.5'
!!_;-8;'12'.;#5#'671.5'$#6$#5#,.5'%,'-,8$#%5-,&'"#718-.E`'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
I15-:1,G'K'
B
A
C
D
3N'
Average vs. Instantaneous Acceleration!
!!P#718-.E'-5'%'8;%,&#'-,'0-567%8#+#,.'>-.;'$#56#8.'.1':+#'
•! R-+-7%$7EG'8;%,&#'-,'"#718-.E'>-.;'$#56#8.'.1':+#'-5'
8%77#0'%88#7#$%:1,'S%'"#8.1$T'
!
Mean Acceleration a "change in velocity
time elasped=#! v
#t=
! v $! v 0
t $ t0
!
Acceleration ! a " lim
#t$0
#! v
#t=
d! v
dt
Putting this together with displacement vector:!
!
Acceleration ! a "
d! v
dt=
d
dt
d! x
dt
#
$ %
&
' ( =
d2 ! x
dt2
This means acceleration is the second derivative of displacement!33'
•! Again, it is useful to define “instantaneous acceleration” (for short, “acceleration)!
I$1?7#+<'!88#7#$%:1,'
!!U1,5-0#$'%'d1$+/7%M3'8%$'5#e,&'1f'2$1+'.;#'&$-0'%,0'
+1"-,&'-,'%'5.$%-&;.'7-,#Q'
!!_;%.'-5'.;#'%"#$%&#'%88#7#$%:1,'-2'%J#$'3N'5#81,05'-.'-5'
.$%"#77-,&'%.'^a+5M3'SgO\N'=+X;T`'
!Q!̂ Qa'+5M3'
YQ!̂ Qa'+5MO'
UQ!NQ3\c'+5M3'
4Q!NQ3\c'+5MO'
,*-N'+5M3'
$*-N+5'
,*-^a++5M3'
$-3N'5'
3O'
I$1?7#+<'L,.#$6$#:,&'I71.5'
!!_;-8;'12'.;#5#'671.5'$#6$#5#,.5'%',#&%:"#'%88#7#$%:1,`'
*-+#G'.'
I15-:1,G'K'
*-+#G'.'
*-+#G'.'
*-+#G'.'
P#718-.EG'"'
B
A
C
D
P#718-.EG'"'
P#718-.EG'"'
3W'
Summary Graphical Analysis of 1D Motion!
x-t, v-t, a-t graphs, and
“motion diagrams”
3\'
Interpreting Displacement!
!$#%'/,0#$'.;#'8/$"#'1,'.;#'$-&;.'Sin this case, constant
aT<'
!!(1>'.;#'%"#$%&#'"#718-.E'-5'<'
*-+#G'.'
P#718-.EG'"'
H.'
H"'
"N'
!
Area = v0"t +
1
2"v"t
!
v =
! v 0
+! v
2=2! v 0"! v 0
+! v
2=! v 0
+
! v "! v 0
2
!
"! v =! v #! v 0
(1>'.;#'8;%,&#'-,'"#718-.E'"#8.1$'-5'<'
L2'>#'+/7:67E'%"#$%&#'"#718-.E'?E'8;%,&#'-,'
:+#<'
!
v "t = Area
3Z'
U%78/7%:,&'4-567%8#+#,.'
!! gradient/slope change, i.e., non-constant acceleration)!To calculate the displacement between A and B we need the area under the curve!
!! ''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''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*-+#G'$+
P#718-.EG',+
A
B
!
Area = ! x
B"! x
A=
! v
t#t
t= tA
tB
$
!
"t
!
lim"t#0
! v
t"t
t= tA
tB
$
!
! v dt
tA
tB
"
!
"! x =
! v dt
tA
tB
#
3^'
!!Now take the limit as "t tends to zero!
U1,5.%,.'!88#7#$%:1,'[D/%:1,5'
!
"! v =
! a dt
tA
tB
#
!
! v =" v 0
+" a t
!
"! v =
! a dt
0
t
# $ ! v %! v
0=! a t
An analogy can be made for velocity vs. acceleration!
Taking constant acceleration (no change in value or !direction)!
Now let’s integrate this!
!
! x "! x 0
=! v dt =
0
t
# (! v 0
+" a t)dt
0
t
#Both v0 and accerlation are constants!
!
! x =! x 0
+! v 0t +
" a t
2
2
"
# $
%
& ' 0
t
=! x 0
+! v 0t +
" a t
2
2 3a'
U1,5.%,.'!88#7#$%:1,'[D/%:1,5'
!
! v =" v 0
+" a t
!
! a =
" v "! v 0
t
!
! x =! x 0
+! v 0t +
" a t
2
2Substitute into!
!
=! x 0
+! v 0t +(v " v
0)t
2
!
=! x 0
+(! v +! v 0)
2t
This states that the displacement vector between starting and ending points is simply the average of the velocity vectors multiplied by elapsed time.!
Similarly, we can use substitution to remove t:!
!
! v 2
=! v 0
2" 2! a (! x "! x 0)
3b'
I$1?7#+<'"g"Nh%.'
!!!'5+%$.'8%$'81+#5'1,.1'.;#'U%7&%$E'*$%-7'.$%"#77-,&'%.'
3a+X5'%,0'%88#7#$%.#5'%.'NQZ'+5MO'21$'WN'5#81,05Q'_;%.'
-5'-.5'B,%7'"#718-.E`'
!Q!O'+X5'
YQ!O'+X5O'
UQ!3aQZ'+X5'
4Q!WO'+X5'
[Q!WO'+X5O'
3c'
Summary!
!!_#',1>';%"#'%'5#.'12'\'#D/%:1,5'.;%.'8%,'?#'%667-#0'.1'
6$1?7#+5'-,"17"-,&'81,5.%,.'%88#7#$%:1,'
!!Selecting the correct one to use requires some care!!! Y#5.'5.$%.#&E'-5'.1'>$-.#'01>,'>;%.'D/%,::#5'%$#'&-"#,'%,0'
>;-8;'+/5.'?#'8%78/7%.#0'
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