Momentum and Collisions in Physics
L-8 (M-7)
I. Collisions II. Work and Energy
•
Momentum:
an object of mass m, moving with
velocity v has a momentum
p = m v
.
•
Momentum is an important and useful concept
that is used to analyze collisions
–
The colliding objects
exert strong forces
on
each other over relatively
short time intervals
–
Details of the forces are usually not known, but
the forces acting on the objects are
equal in
magnitude and opposite in direction (3
rd
law)
–
The law of conservation of momentum which
follows from Newton’s 2
nd
and 3
rd
laws, allows
us to predict what happens in collisions
1
2
I. Physics of collisions:
conservation of momentum
•
The concept of momentum is very useful
when discussing how 2 objects interact.
•
Suppose two objects are on a collision
course.
A
B
•
We know their masses and speeds before
they collide
•
The momentum concept helps us to
predict what will happen after they collide.
3
Law of Conservation of Momentum
•
A consequence of Newton’s 3
rd
law is that if we add the
momentum of both objects before
a collision, it is the same as the
momentum of the two objects
immediately
after the collision.
The collision redistributes the
momentum among the objects.
•
The law of
conservation of
momentum
and the law of
conservation of energy
are two of
the fundamental laws of nature.
During the short time of
the collision, the effect of
gravity is not important.
4
Newton’s Cradle
Momentum conservation in a two-body collision,
How it works.
5
Example: big fish eats little fish
A big fish, M = 5 kg swimming at 1 m/s eats a little
fish, m = 1 kg that is at rest. What is the speed of
the big fish just after eating the little fish?
•
The two fishes form a
system
and their momentum before
the “interaction” is the same as their momentum after the
“interaction”.
•
Momentum before
= M V
before
+ m (0) = 5 kg x 1 m/s
•
Momentum after
= (M + m) V
after
= (5 + 1) V
after
•
5 kg m/s = 6
V
after
V
after
= 5/6 m/s
6
Energy considerations in collisions
•
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:
K
E
=
½
m
v
2
(
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.
.
•
In the collision of two moving objects, both have KE
•
As a result of the collision, the KE of the objects
may decrease because the objects get damaged,
some heat is produced as well as sound.
•
Only if the objects bounce off of each other
perfectly, with no permanent damage
(perfectly
elastic)
is the KE conserved. “
Real” collisions are
never perfectly elastic.
7
Types of collisions
•
Elastic collision
: the two objects bounce off
each other with no loss of
energy
.
•
Inelastic collision
:
the two objects bounce
off each other but with some loss of
energy
.
Most realistic (everyday) collisions are of
this type.
•
Completely inelastic collision
:
The two
objects stick together after the collision.
This type of collision involves the largest
possible loss of
energy
.
8
“Super balls” make almost perfectly
elastic collisions
•
A perfectly elastic
“super ball”
rebounds to
the same height after
bouncing off the floor; it
leaves the floor with the
same KE it had before it
hit the floor
•
A
“real” ball
(not perfectly
elastic) does not return
to the same height;
some of its KE is lost
9
Perfectly elastic collision
m
m
v
before
m
m
v
after
momentum before = m v, KE
before
= ½ mv
2
momentum after = m v, KE
after
= ½ mv
2
Both momentum and KE are conserved
10
v = 0
C
o
m
p
l
e
t
e
l
y
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K
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i
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r
v
e
d
m
m
v
B
E
F
O
R
E
m
m
A
F
T
E
R
momentum before = m v + m 0 = m v
momentum after = (2 m) v/2 = m v
2m
v = 0
11
½ v
KE before = ½ mv
2
KE after = ½ (2m)(v/2)
2
=1/4 mv
2
= ½ KE before (half of the original KE is lost)
Football: a game of collisions
Football players exert
equal forces on each
other in opposite
directions
12
Sumo wrestling
13
non-violent “collisions”
•
Two stationary ice skaters push off
•
both skaters exert equal forces on each other
•
however, the smaller skater acquires a larger
speed than the larger skater.
•
momentum is conserved!
14
S
e
e
Y
o
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T
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b
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f
o
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m
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f
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f
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S
h
o
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t
i
n
g
15
R
E
C
O
I
L
Recoil
•
That “kick” you experience when you fire a
gun is due to conservation of momentum
•
Before firing the cannon
its
momentum = 0
•
Conservation of momentum requires that
after the cannon is fired the total (cannon
plus ball) momentum must still be zero
16
Recoil of a cannon
•
Cannon mass M, velocity V; ball mass m, velocity v
•
The system (cannon and ball) are initially at rest so
the initial momentum = 0
•
The momentum remains 0 after the ball is fired, so
the final momentum = MV + mv = 0
•
The recoil velocity of the cannon in then: V = -mv/M
•
V is in the opposite direction to the ball and much
less then the speed of the ball, v
17
Recoil propels rockets
hot gas ejected at
very high speed
18
II. Work and Energy
•
These terms have a
common meaning
in
everyday usage which may not be the
same as the
physics definitions
•
If we have
“energy”
we can do things:
perform
work
(useful)
•
Energy is the ability to do work
•
We must give precise definitions to work
and energy
•
We have already seen that objects in
motion have
KE = ½ mv
2
19
Work and energy
•
According to the physics
definition, you are NOT
doing work if you are just
holding the weight above
your head
•
you are doing work only
while you are
lifting
the
weight above your head
•
In physics,
WORK requires
both force and motion in the
direction of the force
20
Work requires:
(a) force and (b) motion (displacement) in
the direction that the force acts
•
W
o
r
k
W
=
f
o
r
c
e
(
F
)
x
d
i
s
p
l
a
c
e
m
e
n
t
(
s
)
:
W
F
=
F
s
•
Unit of work:
–
force (N) x distance (m) = N m
–
1 N m = 1 J (Joule)
•
Gravity, mg also acts on the box but does
NO work because there is no vertical motion
21
F
o
r
c
e
,
F
displacement, s
m
g
Physics definition of WORK
•
to do work on an object you have to push
the object a certain distance in the
direction that you are pushing
•
W
o
r
k
=
f
o
r
c
e
x
d
i
s
p
l
a
c
e
m
e
n
t
=
F
s
•
If I carry a box across the room I do not do
work on it because the force is not in the
direction of the motion
22
Who’s doin the work
around here?
NO WORK
WORK
23
A ramp is actually a machine
•
A
m
a
c
h
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a
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p
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t
a
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k
m
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r
e
e
a
s
i
l
y
•
it does not need to have any moving parts.
WORK DONE
= big force
little distance or little force
big distance
24
A lifting machine: Block and tackle
25
Momentum plays a crucial role in analyzing collisions, where objects exert forces on each other over short time intervals. Conservation of momentum, following Newton's laws, allows predicting outcomes in collisions by redistributing momentum among objects. The concept is illustrated through examples like the Newton's Cradle and the scenario of a big fish eating a small fish. Energy considerations also come into play in collision scenarios.
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Presentation Transcript
L-8 (M-7) I. Collisions II. Work and Energy Momentum: an object of mass m, moving with velocity v has a momentum p = m v. Momentum is an important and useful concept that is used to analyze collisions The colliding objects exert strong forces on each other over relatively short time intervals Details of the forces are usually not known, but the forces acting on the objects are equal in magnitude and opposite in direction (3rdlaw) The law of conservation of momentum which follows from Newton s 2ndand 3rdlaws, allows us to predict what happens in collisions 1
I. Physics of collisions: conservation of momentum The concept of momentum is very useful when discussing how 2 objects interact. Suppose two objects are on a collision course. A B We know their masses and speeds before they collide The momentum concept helps us to predict what will happen after they collide. 3
Law of Conservation of Momentum Newton s Cradle A consequence of Newton s 3rd law is that if we add the momentum of both objects before a collision, it is the same as the momentum of the two objects immediately after the collision. The collision redistributes the momentum among the objects. The law of conservation of momentum and the law of conservation of energy are two of the fundamental laws of nature. During the short time of the collision, the effect of gravity is not important. 4
Momentum conservation in a two-body collision, How it works. vB, before vA, before before collision B A vB, after vA, after after collision B A ( m v ) ( ) m v p + p = p + p A B A = B before collision + m v after collision + m v A A, before B B, before A A, after B B, after 5
Example: big fish eats little fish A big fish, M = 5 kg swimming at 1 m/s eats a little fish, m = 1 kg that is at rest. What is the speed of the big fish just after eating the little fish? The two fishes form a system and their momentum before the interaction is the same as their momentum after the interaction . Momentum before = M Vbefore + m (0) = 5 kg x 1 m/s Momentum after = (M + m) Vafter = (5 + 1) Vafter 5 kg m/s = 6 Vafter Vafter = 5/6 m/s 6
Energy considerations in collisions Objects that are in motion have kinetic energy: KE = m v2(Note that KE does not depend on the direction of the object s motion) more on this . . . In the collision of two moving objects, both have KE As a result of the collision, the KE of the objects may decrease because the objects get damaged, some heat is produced as well as sound. Only if the objects bounce off of each other perfectly, with no permanent damage (perfectly elastic) is the KE conserved. Real collisions are never perfectly elastic. 7
Types of collisions Elastic collision: the two objects bounce off each other with no loss of energy. Inelastic collision: the two objects bounce off each other but with some loss of energy. Most realistic (everyday) collisions are of this type. Completely inelastic collision: The two objects stick together after the collision. This type of collision involves the largest possible loss of energy. 8
Super balls make almost perfectly elastic collisions A perfectly elastic super ball rebounds to the same height after bouncing off the floor; it leaves the floor with the same KE it had before it hit the floor A real ball (not perfectly elastic) does not return to the same height; some of its KE is lost 9
Perfectly elastic collision v = 0 v m before m v after m m momentum before = m v, KEbefore = mv2 momentum after = m v, KEafter = mv2 Both momentum and KE are conserved10
Completely inelastic collision: objects stick together momentum is conserved but KE is not conserved BEFORE AFTER v = 0 2m v m m v m m momentum before = m v + m 0 = m v momentum after = (2 m) v/2 = m v KE before = mv2 KE after = (2m)(v/2)2 =1/4 mv2 = KE before (half of the original KE is lost) 11
Football: a game of collisions Football players exert equal forces on each other in opposite directions 12
non-violent collisions Two stationary ice skaters push off both skaters exert equal forces on each other however, the smaller skater acquires a larger speed than the larger skater. momentum is conserved! 14
RECOIL See You Tube for more videos of Rifle Shooting 15
Recoil That kick you experience when you fire a gun is due to conservation of momentum Before firing the cannon its momentum = 0 Conservation of momentum requires that after the cannon is fired the total (cannon plus ball) momentum must still be zero 16
Recoil of a cannon Cannon mass M, velocity V; ball mass m, velocity v The system (cannon and ball) are initially at rest so the initial momentum = 0 The momentum remains 0 after the ball is fired, so the final momentum = MV + mv = 0 The recoil velocity of the cannon in then: V = -mv/M V is in the opposite direction to the ball and much less then the speed of the ball, v 17
Recoil propels rockets hot gas ejected at very high speed 18
II. Work and Energy These terms have a common meaning in everyday usage which may not be the same as the physics definitions If we have energy we can do things: perform work (useful) Energy is the ability to do work We must give precise definitions to work and energy We have already seen that objects in motion have KE = mv2 19
Work and energy According to the physics definition, you are NOT doing work if you are just holding the weight above your head you are doing work only while you are lifting the weight above your head In physics, WORK requires both force and motion in the direction of the force 20
Work requires: (a) force and (b) motion (displacement) in the direction that the force acts displacement, s Force, F mg Work W = force (F) x displacement (s): WF = F s Unit of work: force (N) x distance (m) = N m 1 N m = 1 J (Joule) Gravity, mg also acts on the box but does NO work because there is no vertical motion 21
Physics definition of WORK to do work on an object you have to push the object a certain distance in the direction that you are pushing Work = force x displacement = F s If I carry a box across the room I do not do work on it because the force is not in the direction of the motion 22
Whos doin the work around here? WORK NO WORK 23
A ramp is actually a machine A machine is any device that allows us to accomplish a task more easily it does not need to have any moving parts. WORK DONE = big force little distance or little force big distance 24
