Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 1
Experimental Physics EP1 MECHANICS
- System of Particles -- Conservation of Momentum -
https://bloch.physgeo.uni-leipzig.de/amr/
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 2
Center of mass
m mm mm m
3m 3m
4x1x 2x 3x 5x 6x 7x å å=i i
iiicm xmmX
471=
+×+×
=mmmmXcm
44
7621=
+++=cmX
233211 =
++=cmX 6
37652 =
++=cmX 4
6)62(3, =
+==
åå
mm
MXM
Xi
icmicm
m1 m3m2
21 mm +å å=i i
iiicm rmmR !!
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 3
3mm
Center of mass
å å === cmiiii MghhmgghmU
grF!
å= iighmU0
å D+= ii hgmUU 0q
qq sin20 mgUU -=
0cos2 =-= qq
mgddU
2m
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 4
R
x
y
Continuous bodiesdm
ò
ò
ò
ò== L
L
L
L
cm
rdr
rdrr
dm
dmrR
0
0
0
0
)(
)(
!!
!!!!!
r
r0 L
221
0
20
0 0
0 0 LLL
dx
xdxX L
L
cm ===òò
rr
r
r
6)(
)( R
xdm
xxdmX R
R
R
Rcm -==
òò-
-
62/)(0 R
mMRmMXcm -=
-×-+×
=
Negative mass:
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 5
Motion of the center of mass
cmR!
extF!
å=i
iicm rmMR !!
å=i
ii
cm
dtrdm
dtRdM
!!
å=i
ii
cm
dtrdm
dtRdM 2
2
2
2 !!
( )åå +==i
iii
icm FFFAM ext,,int
!!!!
extA!
cmAMF!!
=net,ext
Under influence of an external force thecenter of mass of a system moves like a pointparticle with a mass equal to the total mass ofthe system and irrespective of internal forcesacting within the system.
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 6
Conservation of momentum
vmp !!=
1v!
net)( Fam
dtvmd
dtpd !!
!!===
Linear momentum:
12F! 21F!
2112 FF!!
-=
dtpd
dtvmdF 22
12)( !!!==
constppdtpd
dtpd
=+Þ-= 2112 !!!!
å å === cmiii VMvmpP!!!!
net,extFdtVdM
dtPd cm
!!!
==
The law of conservation of momentum:
constVMvmP cmii ===å!"!
2v!
interaction
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 7
0 5 10 15 20 25 30 35 40 45
-8
-6
-4
-2
0
2
4
6
8 tv!
0v!
y
x
q0
Projectile with explosion
22
0000 )cos(2)(tan x
vgxyyq
q -+=
x
toptop v
xxgyy
2
2
2 2)( -
-=
m2
m
m
2102 vmvmvm !!!+=
xx mvmv 202 =
xx vv 02 2=
?
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 8
0 5 10 15 20 25 30 35 40 45
-8
-6
-4
-2
0
2
4
6
8 tv!
0v!
y
x
q0
Projectile with explosion
22
0000 )cos(2)(tan x
vgxyyq
q -+=
x
toptop v
xxgyy
2
2
2 2)( -
-=
m2
m
m
2102 vmvmvm !!!+=
xx mvmv 202 =
xx vv 02 2=
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 9
Rocket propulsionv
0M
gu
constvmP ii ==å !!
ug is relative to the rocket;
dtdmR =
At an arbitrary time t
( ) ( )( )dvvdmmdmuvmv g +-+-=
At a time t+dt
gg Rudtdmu
dtdvm == - the thrust
ò -=-=t
RtMdmMm0 00
( )òò --=t
g
v
vdtRtMRudvf
i 0
10
extF!
+
÷÷ø
öççè
æ-
+=RtM
Muvv gif0
0ln
0 5 10 15 20 25 30 35 40 45 50 55
0
10
20
30
40
spee
d, m
/s
time, s
R is the rate at whichgas is exhausted
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 10
m1
Skaters on ice
m2
m1 m21v!
2v!
0=å ivm!
2211 vmvm =
tx
mtx
m 22
11 =
extF!
cmcmext VAmF Þ=!!
( ) ( ) cmcmcm VmmVvmVvm )( 212211 +-=-+--
cmVvu!!!
+= 02211 =+- vmvm
( ) åååå å ++=+== iicmcmiiicmiiiik vmVVmvmVvmumE 2222
21
21
21
21
rel,2
21
kcmk EMVE += Ek,rel is reference-independent
Experimental Physics - Mechanics - System of Particles - Conservation of Momentum 11
Ø The center of mass of a system moves like a point-like
object having the total system mass.
Ø If there is no net force acting upon the system, then
the linear momentum is conserved.
Ø The total kinetic energy of the system can be written
as the sum of the kinetic energy associated
with the center of mass plus the kinetic
energy associated with the particle
movements in the center-of-mass
reference system.
To remember!
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 12
Elastic collision
constP
constE
total
totalk
=
=,
))(())(( 1111122222 fifiifif vvvvmvvvvm +-=+-
2222
12112
12222
12112
1iiff vmvmvmvm +=+constE totalk =,
constPtotal = iiff vmvmvmvm 22112211 +=+
)()( 111222 fiif vvmvvm -=-
Þ+=+ fiif vvvv 1122 )( 1212 iiff vvvv --=-
Relative speed of approach is equal to relative speed of recession.
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 13
Center-of-mass reference frame
constVMvmP cmii ===å!"!
icm uV 1+ icm uV 2+
ii pp 21 -=
÷÷ø
öççè
æ+=+=+=
21
21
2
22
1
21
222
211
, 21
21
2222 mmp
mp
mpvmvmE i
iiiiik
÷÷ø
öççè
æ+=+=+=
21
21
2
22
1
21
222
211
, 21
21
2222 mmp
mp
mpvmvm
E fffff
fk
No external forces!
Elastic collision Þ Þ21
21 fi pp =
ïî
ïíì
±=
±=
if
if
pp
pp
22
11
ïî
ïíì
±=
±=
if
if
uu
uu
22
11
In CoM reference frame the velocities of each object are simply reversed by collision.
Experimental Physics - Mechanics - Selected problems 14
Astroblaster
0v
0v0v
0v
2m
1m2v
1v
22110201 vmvmvmvm +=-
)( 1212 iiff vvvv --=-
00012 2)( vvvvv =---=-
02v
0v3v3m
2m
332203022 vmvmvmvm +=-
00023 3)2( vvvvv =---=-
021
211
3 vmmmmv
+-
=021
21
2;03
vvvmm
===
32
3202
22mmmmvv
+-
=
032
32
3;02
vvvmm==
=
( ) 920303 == vvhh
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 15
Perfectly inelastic collision
iv1 iv2 cmV
cmii MVvm =å
2
22
1
21
222
211
, 2222 mp
mpvmvmE iiii
ik +=+=)(22 21
2221
, mmpVmmE cmfk +
=+
=
02 =iv2)(2
2 211
21
1
21
,1,
iikfk
vmmm
mmmEm
E+
=+
=21
21
mmvv ii
=-= 0, =fkE
bv hghmmvm
mmmE i
fk )(2 21
211
21
1, +=
+=
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 16
Coefficient of restitution
)( 1212 iiff vvevv -×-=-e = 1 – elastic collisione = 0 – perfectly inelastic collision
1h2h1v
2v
120 evvvplate =Þ=
2
1
2
1
1
2
hh
mghmgh
vve ===
Table tennisStainless tablee = 0.92
Bat-bat COR e < 0.5
?)( -=D efEk MpE ik
2
, = MpeE fk
22
, = )1( 2, -=D eEE ikk
ò=f
i
t
tdtFI!!
- impulse ò -==f
i
t
t if ppdtdtpdI !!!!
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 17
Inelastic collision in 2D
cmii VMvmP!!!
==å
1p!
2p!
°= 30q km/h401 =v
km/h70tan
12 ==
qvv
P!
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 18
Elastic collisions in 2D
2q
m2
m1iv1!
m1
02 =iv!
fv1!
fv2!
1q
constvmP ii ==å !!
îíì
+==
+==
2,221,11
2,221,11,11
sinsin0
coscos
fxfxy
fxfxixx
vmvmP
vmvmvmP1
2
3 2222
12112
12112
1ffi vmvmvm +=
21 mm = 22
21
21 ffi vvv +=ffi vvv 211
!!!+=
iv1!
fv2!
fv1!
Experimental Physics - Mechanics - Elastic and Inelastic Collisions 19
Ø During inelastic collisions the total kinetic energy is not
conserved. It is conserved during elastic collision.
Ø In a perfectly inelastic collision the bodies stick together and
move with the velocity of COM.
Ø In the COM reference frame, upon 1D elastic collision
the bodies change their velocities to
the opposite ones.
Ø For non-central collisions one has to
consider conservation of all components
of the linear momentum.
To remember!
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