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System of particles,
Linear momentum, collision,
Lecture 2
1. Linear momentum and collision
Def. : Linear momentum vmp
t
p
t
vmamF
dt
Def. : Impulse )(tp)(tp(t)dtFI 12
t
t
2
1
Average force:
12
12ave
tt
)(tp)(tp
t
IF
)(tp)(tptFI 12ave
Examples!!!
Conservation of linear momentum I.
internal
forces
1F
2F
external
forces
2211211221net amamFFFFF
Newton III.
11211 amFF
22122 amFF
netext.,F
t
vm
t
vmF 2
21
1netext.,
m1=const.
m2=const.
t
)vm(
t
)vm(F 2211
netext.,
t
)p(
t
)p(F 21
netext.,
t
)pp(F 21
netext.,
t
)p(F
syst.
netext.,
Conservation of linear momentum II.
If 0F ext. net,
i
isyst. const.pp
Def. : Center of mass (center of gravity)
ii
iii
cm
rm
r
Velocity of the center of mass
(of system of particles) M
p
m
vm
t
rv
syst.
ii
iii
cc
Acceleration of the center of mass t
p
M
1
t
va
syst.cc
M
F
m
F
t
va
net ext,
ii
ii ext,
cc
t
)p(F
syst.
netext.,
t
p
M
1
t
va
syst.cc
If 0F ext. net,
const.vc
const.ac
If 0F ext. net,
0vc
& const.rc
Conservation of linear momentum III.
Example!!!
Completely inelastic collision
m1
m2
1v
2v
v
vmmvmvm 212211
Conservation of linear momentum
Energy dissipation:
2
22211
221k vm
2
1vm
2
1vmm
2
1ΔE
(Work of deformation, transfer to heat, etc…)
Completely inelastic collision: an example → ballistic pendulum
m
m+M
M
φ
Conservation of linear momentum:
)cos-2gL(1m
MmvA
v
0vB
ghMmvMm2
1 2 After the collision:
)cos-2gL(1gh2v 2
vMmvm A
Elastic collision
m1
m2
1v
2v
1v
2v
Conservation of linear momentum:
22112211 vmvmvmvm
Conservation of energy:
222
211
222
211 vm
2
1vm
2
1vm
2
1vm
2
1
Example:
v2=0, m1=m2
Example:
v2 = 0, m1 = m2
Elastic collision in 1D
”Stress reliever”
Collision in 2D
Conservation of linear momentum:
Example:
x:
y:
If the collision is elastic:
Completely inelastic collision in 2D: example → collision of cars
vmmvmvm 212211
1p
2p
fp
1p
2p fp
Rocket propulsion
