!"#$%&'#(
)*$(+,%$-.(#"#$%&'#("/(-(0"12(,3
(1$4%$1(-35(
!!
momentum
! = mass " velocity
!
P = mv
!!
!
#"#$%&'#(,3(-(6$7&".(8-.-++$+(&"(&*$(6$+"7,&2
(6$7&".(
)*$('%,&3("/(#"#$%&'#(7-%(3,#8+2(0$
(1$&$.#,%$1(/."#(&*$(1$4%,9"%(-3(:;#3<=(
!!
!
!!
!-33(>:;?(@(A$+"7,&2(>#3<=?(
!"#$%&'#(,3(.$-++2(-(#$-3'.$("/(&*$
(BC'-%9&2("/(#"9"%D(&*-&(-%("0E$7&(*-3(
!!
!!
)*$(#".$(#"#$%&'#(-%("0E$7&(*-3(&*$
(*-.1$.(,&(,3(&"(3&"8(,&3(#"9"%(
=(:;(
=R(#S3$7(
!"#!"#$$%&'(#)!*'!+,-%'!"$%&
"'(#&)*)#+,')&-"#+&"#&.'+%/&
01(&)$%%&1%&2$.,&*3&1,4&
=RRR(:;(
OR=(#S3$7(
B$)#&C*)#+,')&
F".7$(-%1(!"#$%&'#(
G$&3(.$&'.%(&"(H$I&"%J3(3$7"%1(+-I( !
!
!
dv d(mv )
F = ma = m =
dt
dt
!!
Therefore, Force is the derivative of Momentum:
!
H"&(&"(0$(7"%/'3$1(I,&*(.$+-9"%(
!
dp
dU
F =
0$&I$$%(K"%3$.6-96$(F".7$(L( F = ! dt !
dx
M"&$%9-+(N%$.;25&
!! ,O$O()*$(.-&$("/(7*-%;$("/(#"#$%&'#(,3(8."8".9"%-+(&"(&*$(/".7$
(>$C'-+(I,&*(PQ('%,&3?(
56&7#-,*+&88&9&
!
!
!"1%&1%&,"#&*.1(1+$:&3*.)&*3&7#-,*+&88/&$+;&1+&3$<,&1%&)*.#&(#+#.$:&,"$+
&&
&&&&&&&&&&&&&&&&&&&&&&&&&%1+<#&1,&.#)$1+%&=$:1;&#=#+&-"#+&,"#&)$%%&!"&=$.1$0:#&>$%&
1+&.*<?#,&2.*2':%1*+@A&&
D! !"3&("T$%(#-33(1"$3%U&(7*-%;$(3"(6$+"7,&2(7*-%;$3(
<<<<<V(%"%W$."(-77$+$.-9"%O((X'&(#-33(7-%(7*-%;$Y(
!
!
dp d
dm
dv
" Fext = dt = dt (mv) = dt v + m dt
EF$)2:#%9&&&&&
motion of a rocket
astroid entering Earth’s atmosphere
X"Z"#(+,%$5(()*$(/".7$(F(,3(*-.1(&"(C'-%9/2[(%$$1(%$I(#$-3'.$(
!!
Q&(-7&3(/".(-(6$.2(3*".&(9#$(>\#3?(
! Q&(,3(%"&(-(7"%3&-%&(/".7$(3,%7$(,&(1$8$%13("%(*"I(1$/".#$1(&*$("0E$7&3(-.$(-&(
!!
&*-&(,%3&-%&(
8)2':%#&$+;&,"#&8)2':%#G)*)#+,')&!"#*.#)&
H*)2$.1+(& &&&&&&&&&&&&&&&&&&&&&&&&&&&-1,"
-"$,I%&,"#&;1J#.#+<#K&
&&&&&&&&&&&&&&&&&&&&&&0*,"&*3&-"1<"&;#2#+;&*+&)&$+;&=/&
L#M+1N*+9&&B'22*%#&-#&"$=#&$&<*+%,$+,&+#,&3*.<#&&&& &&&&&
$<N+(&*+&$&2$.N<:#&;'.1+(&$&N)#&1+,#.=$:&O,P,QG,RA&&S#&;#M+#&
,"#&1)2':%#&$%&,"#&2.*;'<,&*3&,"#&+#,&3*.<#&$+;&,"#&N)#&
H*)2$.1%*+9& !
1+,#.=$:9&
!
W total = " F • #x
(
8)2':%#G)*)#+,')&,"#*.#)&3*.&<*+%,$+,&3*.<#%9&
839&
!"#+9&
!
T#+#.$:&8)2':%#G)*)#+,')&,"#*.#)9&
)
Notes on Impulse!
]*.<#/&]&
!"#&3*.<#ON)#&.#:$N*+%"12&1%&*0%#.=#;&1+&)$+6&U.#$:&-*.:;V&
#F$)2:#%9&
W! ]'&"#"0,+$(3-/$&2(
D! L$%"0*$.;%&
D! X1.0$(%&
D! H.')2:#&Y*+#%&>,"#&;1%N+<,&$;=$+,$(#&*3&6*'.&,.'%,#;&
H"#=6&*=#.&B)$.,&<$.@&
W! M."1'7&(8-7:-;,%;(
D! B,6.*3*$)&%2$<#.%&
W! P8".&3(
D! !#++1%9&.$<?#,&%,.1+(&,#+%1*+&
D! 5$%#0$::9&UZ'1<#;V&0$%#0$::%&[&0$%#0$::&0$,%&><*.?#;&[&
$:')1+')&=%A&-**;@&
D! T*:39&,"#&U%2.1+(G:1?#V&#J#<,&*3&(*:3&<:'0&"#$;%&
D! 5*F1+(&(:*=#%9&>:*-#.&3*.<#%&1+&,"#&"$+;%&[&3$<#@&
!
!
!
I = Area = lim $ Fi"t % I =
"t#>0
t
&
t2
t1
!
Fdt
AAA&
\$.(#%,&8)2':%#K&
!1)#/&,&
^*$%(7.-3*(&$39%;(-(%$I(7-.(,1$%97-+(6$*,7+$3(-.$
(1.,6$%(-&(&*$(3-#$(38$$1(,%&"(-(*-.1$%$1(7"%7.$&$
(I-++[(-(&.$$(-%1(-%($@&.$#$+2(&*,7:(/"-#(.'00$.(0+"7:O
(Q%($-7*(7-3$(&*$(7-.($%13('8(-&(.$3&O(^*,7*(7-3$(I,++
(8."1'7$(&*$(+-.;$3&(,#8'+3$_(
!!
!
]O!F"-#(.'00$.(
XO!).$$(
KO!K"%7.$&$(I-++(
`O!)*$(7"%7.$&$(I-++(-%1(&.$$($C'-++2(
NO!)*$2(-++(8."1'7$(&*$(3-#$(,#8'+3$(
! ! !
I = p2 " p1
Enough said
!
\$.(#%,&8)2':%#K&
^*,7*("/(&*$(/"++"I,%;(,%6"+6$3(&*$(+-.;$3&(,#8'+3$_(
!!
]O!(](7"%3&-%&(aH(/".7$(-79%;(/".(b3(
XO!(](/".7$(I*,7*(,%7.$-3$3(+,%$-.+2(/."#(R(&"(=RH("6$.(-(8$.,"1
("/(c3(
KO!(](/".7$(I*,7*(1$7.$-3$3(+,%$-.+2(/."#(=RH(&"(RH("6$.((8$.,"1("/(a3(
`O!(](/".7$(I*,7*(3&-.&3(-&(RH[(+,%$-.+2(,%7.$-3$3(&"(=RH(-%1(&*$%
(+,%$-.+2(1$7.$-3$3(&"(RH("6$.(-(&"&-+(8$.,"1("/(d3(
NO!)*$2(-++(*-6$(&*$(3-#$(,#8'+3$(
F
F
R^&
t
F
R_&
t
F
R^&
t
Q^&
t
M*23,73("/(^-&$.(e"7:$&3(
>-O:O-5((./#(0%)1!(*&!&234%)1!5267%!&234%)8?((
H*)2.#%%#;&
X1.&
B##&$+1)$N*+&
8+1N$:&
S$,#.&
#(*+,'&)
]1+$:&
#$%&'()
X&.*<?#,&'%#%&<*)2.#%%#;&$1.&,*&
3*.<#&-$,#.&*',&*3&$&+*YY:#&
S$,#.&
-$,#.&.*<?#,%&
S$,#.&`*<?#,&
)*$(7*-%;$(,%(#"#$%&'#("/(&*$(I-&$.(
"p = mwaterv water
H"I(/".(+-.;$(>#-7."37"8,7?(7*-%;$35( F =
"p
"t
P"(-(/".7$(#'3&(-7&("%(&*$(I-&$.(&"(7*-%;$(,&3(#"#$%&'#(
!!
!!
)*$(/".7$(,3(7-'3$1(02(&*$(7"#8.$33$1(-,.(
)*$(#-;%,&'1$("/(&*$(/".7$(,3(,%6$.3$+2(8."8".9"%-+(&"(&*$(9#$(/".
(I*,7*(,&(-7&3(
!
!
H$I&"%J3(&*,.1(+-I(.$C',.$3(&*-&(&*$.$(0$(-%($C'-+(-%1
("88"3,&$(/".7$("%(&*$(."7:$&(
)*,3(,3(*"I(-++(."7:$&3(I".:[(%"&(E'3&(&*$(I-&$.(8"I$.$1
("%$3(>3"..2[(`.O(N6,+U3(38-7$3*,8(,3(%"&(
3*"I%(1'$(&"(e(.-9%;Y?(
X1.&`*<?#,K&
K"'+1(I$('3$(-,.(,%(8+-7$("/(I-&$._(
!!Q/(&*$(6"+'#$("/(&*$(."7:$&(,3(A(&*$%(&*$(#-33("/(I-&$.(,3
"pwater = # waterV v
mwater = " waterV
(3,#8+2(
!!
!!
H"I(,/(-,.(I-3('3$1(,%3&$-1(&*$%(>-33'#$(I-&$.(-%1(-,.(&-:$3
(&*$(3-#$(9#$(&"(3C',.&("'&("/(0"Z+$?5(
"pair = # airV v
mair = " airVair
!
!
!!.-9"("/(&*$(&*.'3&(,%(&*$(&I"(7-3$3(,3OOO(
Fwater "pwater /"t "pwater # waterVv 1000
=
=
=
=
= 775
! Fair
"pair /"t
"p
#
Vv
1.29
! air
air
e$#-.:5&&&&
!
&>=?((X2('3,%;(I-&$.(,%(&*$(."7:$&(,%3&$-1("/(-,.(&*$(&*.'3&(,3(,%7.$-3$1(02(-(/-7&".("/(
ffb(>,O$O[(2"'.(g-,.U(."7:$&(,3(XheQHi(-%1(I,++(%"&(,#8.$33Y?(
>a?(^-&$.(-%1(-,.(I,++(-7&'-++2(Hh)($@,&(-&(&*$(3-#$(38$$1(1'$(&"(1,j$.$%7$3(,%(
6,37"3,9$3(>B397:,%$33k?(<<<<(V((-,.(;$&3("'&(/-3&$.(1'$(&"(+"I(6,37"3,&2(
Linear Momentum Conservation!
H$I&"%J3(3$7"%1(+-I(
!!
!!
So when no net force acts there is no change in
momentum
!
!
! !
!
"F = F
12
+ F21 =
#p2 #p1 # !
!
+
= ( p2 + p1 ) = 0
#t
#t #t
`.O(N6,+(-%1(]'39%(M"I$.3(3&-%1("%(,7$O((h%$("/(&*$#
(8'3*$3(&*$("&*$.[((
!!
!!
!
^*-&(,3(&*$(.$+-9"%3*,8(0$&I$$%(&*$,.(4%-+(6$+"7,9$3_(
#-)
#.)
X2('3,%;(&*$(8.,%7,8+$("/(7"%3$.6-9"%("/(#"#$%&'#(
!!
0 = m1v1 '+m2v 2 '= mv1 '+mv 2 '
v1 '= "v 2 '
)*'3(0"&*(3*"'+1(#"6$("j(I,&*(&*$(3-#$(38$$13Y(
!!
!
!
C*)#+,')&H*+%#.=$N*+&
P'88"3,%;("%$(&*$(&I"(,3(%"I(+$-%,%;(-;-,%3&(&*$
(/$%7$(>I-++?[(I*-&(I,++(*-88$%_(
!!
!!
)*$("&*$.(7*-.-7&$.(#"6$3("j(I,&*(3"#$(6$+"7,&2(
#)
Q3(#"#$%&'#(7"%3$.6$1(,%(&*,3(7-3$_(
]O!l$3(
XO!H"(
KO!H"&($%"';*(,%/".#-9"%(&"(1$&$.#,%$(
!!
IF the momentum is conserved:
v1 '= "v 2 ' _( 5',&*+#&*3&,"#)&1%&+*,&)*=1+(/&;*#%&1,&
)#$+&,"#&*,"#.&*+#&1%&+*,&)*=1+(&KK44&&
H*+%,$+,&C*)#+,')&
!"#$%&'#(,3(]G^]lP(7"%3$.6$1OOO0'&(&*$(#"#$%&'#
("/(-(8-.97'+-.(323&$#(,3(%"&(-+I-23(7"%3&-%&(>3-#$(-3
(&"&-+($%$.;2(63O($%$.;2("/(-(323&$#?Y(
!!
!!
Q/(I$(7"%3,1$.(&*$(7-3$("/(&*$(&I,%(0."&*$.3(-3(&*$(323&$#
(&*$%(I$(*-6$(-%($@&$.%-+(/".7$(-79%;(>&*$(I-++(8'3*?(
#)
)*$(#"#$%&'#("/("'.(323&$#(,3(%"&(7"%3&-%&5(&*$
($@&$.%-+(/".7$(-113(#"#$%&'#(&"(:$$8(&*$(g4@$1U(7-.&
(4@$1(,%(8+-7$(
!!
!!
OOO0'&(&*$($C'-+(-%1("88"3,&$(.$-79"%(&"(&*$($@&$.%-+(/".7$
(-113(-%($C'-+(-%1("88"3,&$(#"#$%&'#(&"(&*$(3'.."'%1,%;
($%6,."%#$%&(
F".7$3(0$&I$$%(8-.&3("/(&*$(323&$#(-.$(%"&($@&$.%-+[("%+2(/".7$3(I*"3$($C'-+(-%1(
"88"3,&$(.$-79"%3(1"(%"&(-7&("%(&*$(323&$#(-.$($@&$.%-+&
Impulse and momentum:
](&$%%,3(0-++(>#-33mRO=(:;?(7"#$3(,%(/."#(&*$(.,;*&(-&(dR(#S3(-&(-%
(-%;+$(=R(1$;(0$+"I(*".,W"%&-+O(()*$%("%(-(0-1(3*"&(e-/-(*,&3(,&(-&(cR
(1$;(-0"6$(&*$(*".,W"%&-+(-&(-(38$$1("/(nR(#S3O((^*-&(,3(&*$(,#8'+3$
("/(&*$(%$&(/".7$(-%1(&*$(-6$.-;$(%$&(/".7$[(-33'#,%;(&*$(7"%&-7&
(>7"++,3,"%?(1'.-9"%(,3(ROR=(3$7O((((
P"+'9"%5(()-:$(8"3,96$(o(&"(&*$(e,;*&(-%1(8"3,96$(l(-3(p8O(()*$
(6$+"7,&2(7"#8"%$%&3(0$/".$(>3'037.,8&(=?(-%1(-T$.(>a?(&*$(&$%%,3(0-++
(,3(*,&(-.$5(
v1x = "40 cos(10) = -39.39 m/s v1y = "40 sin(10) = -6.95 m/s
v 2x = 60 cos(30) = 51.96 m/s v 2y = 60 sin(30) = 30 m/s
!
!
o<7"#8"%$%&("/(,#8'+3$(,3($C'-+(&"(@<7"#8"%$%&("/(#"#$%&'#
(7*-%;$(
J = p " p = m(v " v )
x
2x
1x
2x
1x
! (51.96 # (#39.39)) = 9.14 kg $ m/s
= 0.1"
!
J y = p2y " p1y = m(v 2y " v1y )
= 0.1" (30 # (#6.95)) = 3.70 kg $ m/s
X2(6$7&".(-11,9"%(
2
2
J = J x + J y = 9.14 2 + 3.70 2 = 9.86 kg " m/s
!
]6$.-;$(%$&(/".7$5(
!
F=
J 9.86
=
= 986 N
"t 0.01
]+&$.%-96$+2[(I$(7"'+1('3$(7"#8"%$%&3("/(q(&"(4%1(7"#8"%$%&3("/(
(-6$.-;$(%$&(/".7$O(()*$(7"#8"%$%&3("/(-6$.-;$(%$&(/".7$(-.$5(
!
Fx =
J x 9.14
=!
= 914 N
"t 0.01
)*$(-6$.-;$(%$&(/".7$(,35(
2
!
!
Fy =
J y 3.70
=
= 370 N
"t 0.01
2
F = Fx + Fy = 914 2 + 370 2 = 835,396 + 136,900 = 986.05 N
$F '
]6$.-;$(F".7$(`,.$79"%5(
!" = tan#1& y ) = 22.04°
% Fx (
Remarks: (1) Keep signs straight due to vector use (2) Because the ball
was Not at rest initially, average force direction is NOT the same as ball
direction (analogy: projectile motion)
!
Linear Momentum Conservation (1D)!
Initial
!
!
!
!
p1i + p2i = m1v1 + m2v 2
Final
!
!
!
!
!
p
+
p
=
m
v
'
+m
v
!1 f !2 f !1 1 !2 '2
p1i + p2i = p1 f + p2 f
!
!
!
!
m1v1 + m2v 2 = m1v '1 +m2v '2
!
!