Linear Momentum
Lecturer:
Professor Stephen T. Thornton
Reading Quiz
A system of particles is known to have a total
kinetic energy of zero. What can you say about
the total momentum of the system?
A)
B)
C)
D)
momentum of the system is positive
momentum of the system is negative
momentum of the system is zero
you cannot say anything about the
momentum of the system
Reading Quiz
A system of particles is
known to have a total
kinetic energy of zero.
What can you say
about the total
momentum of the
system?
A) momentum of the system is positive
B) momentum of the system is
negative
C) momentum of the system is zero
D) you cannot say anything about the
momentum of the system
Since the total kinetic energy is zero, this means
that all of the particles are at rest (v = 0).
Therefore, since nothing is moving, the total
momentum of the system must also be zero.
Last Time
Conservation of Energy
Escape velocity
Power
Potential energy diagrams
Today
Define linear momentum
Relationship between K.E. and momentum
More general form of 2nd law
Impulse
Internal and external forces
Collisions
New Concept – Linear Momentum
p  mv
Linear momentum is simply the
product of mass and velocity.
Linear momentum is a vector.
Sometimes we say just “momentum”.
SI unit: kg · m/s
2
1 2 1  mv 
K  mv  m 

2
2  m 
2
2
1 p
p
K m 2 
2 m
2m
Kinetic energy and linear momentum are
intimately related. Remember this
result:
2
p
K
2m
Do demo with bouncing ball and
bean bag.
One recoils, the other doesn’t.
Change in Momentum
p  p f  pi
a ) pi  mvˆj
pf  0
p   mvˆj
b) pi  mvˆj p f  mvˆj
p  2mvˆj
Momentum is a vector
ptotal  p1  p2  p3  ...
Momentum and Newton’s 2nd Law
F  F
i
net
 ma
i
A more general form of
Newton’s 2nd law:
dp
Fnet 
dt
Let’s see if the equations are consistent.
p  p f  pi  m f v f  mi vi
 m(v f  vi )  mv , if mass is constant.
p
v
m
 ma
t
t
p dp
so Fnet 

 ma if mass is constant.
t dt
dp
But when mass is not constant, our
Fnet 
new general form should be used.
dt
Impulse
What is impulse and why is it useful?
Forces sometimes act between objects
over very short times.
Examples:
Bouncing balls
Bat hitting a ball
Collisions
The Average Force During a Collision
Definition of Impulse J
Force can vary considerably over the
time of interaction, so let’s consider the
average force, Fav :
J  Fav t
unit is N  s
p
Fav 
t
Fav t  p
tf
and J  Fav t   Fdt  p  J
ti
Impulse is just the change in momentum!
tf
J  Fav t   Fdt  p
ti
Do egg in a sheet demo.
Conservation of Linear Momentum
What happens when Fnet = 0?
Then
dp
Fnet 
dt
p  0 and momentum is conserved!
We have pi  p f
If the net force acting on an object is
zero, its momentum is conserved.
Do demos:
Rocket bicycle
Reaction cars
Fire extinguisher rocket cart
Water rocket
2-liter bottle rocket
Conservation of Linear Momentum
Law of conservation of linear momentum:
When the net external force on a system of
objects is zero, the total momentum of the
system remains constant.
Equivalently,
The total momentum of an isolated system
remains constant.
Conceptual Quiz
An open cart rolls along a
frictionless track while it is
raining. As it rolls, what
happens to the speed of the
cart as the rain collects in it?
(Assume that the rain falls
vertically into the box.)
A) speeds up
B) maintains
constant speed
C) slows down
D) stops immediately
Conceptual Quiz
An open cart rolls along a
frictionless track while it is
raining. As it rolls, what
happens to the speed of the
cart as the rain collects in it?
(Assume that the rain falls
vertically into the box.)
A) speeds up
B) maintains
constant speed
C) slows down
D) stops immediately
Because the rain falls in
vertically, it adds no momentum
to the box, thus the box’s
momentum is conserved.
However, because the mass of
the box slowly increases with the
added rain, its velocity has to
decrease.
Follow-up: What happens to the cart when it stops raining?
Internal and External Forces
If we have a system of particles, then
there can be internal forces (for example,
that hold the object together).
Internal forces always occur in actionreaction pairs and the sum will be zero.
F
int
0
Fnet   Fext
System of Objects
Internal forces have no effect on the net
momentum of an object.
If the net external force acting on a system is
zero, then the net momentum is conserved.
Momentum of every particle in system is not
conserved, only the net.
pnet  p1  p2  p3  ...  constant
Conservation of Momentum
Momentum conservation works for a rocket as long as
we consider the rocket and its fuel to be one system and
account for the mass loss of the rocket.
Rocket Travel. A rocket of total mass
3180 kg is traveling in outer space with a
velocity of 115 m/s. To alter its course
by 35.0°, its rockets can be fired briefly
in a direction perpendicular to its original
motion. If the rocket gases are expelled
at a speed of 1750 m/s, how much mass
must be expelled?
Conceptual Quiz
A system of particles is known to
have a total momentum of zero.
Does it necessarily follow that the
total kinetic energy of the system
is also zero?
A) yes
B) no
Conceptual Quiz
A system of particles is known
to have a total momentum of
A) yes
zero. Does it necessarily follow
B) no
that the total kinetic energy of
the system is also zero?
Momentum is a vector, so the fact that ptot = 0 does
not mean that the particles are at rest! They could be
moving such that their momenta cancel out when you
add up all of the vectors. In that case, because they
are moving, the particles would have non-zero KE.
Conceptual Quiz
Two boxes, one heavier than the
other, are initially at rest on a
horizontal frictionless surface.
The same constant force F acts
on each one for exactly 1 second.
Which box has more momentum
after the force acts ?
F
A) the heavier one
B) the lighter one
C) both the same
light
F
heavy
Conceptual Quiz
Two boxes, one heavier than the
other, are initially at rest on a
horizontal frictionless surface.
The same constant force F acts
on each one for exactly 1 second.
Which box has more momentum
after the force acts ?
We know:
p ,
Fav 
t
so impulse p = Fav t.
In this case F and t are the
same for both boxes!
Both boxes will have the
same final momentum.
F
A) the heavier one
B) the lighter one
C) both the same
light
F
heavy
Conceptual Quiz
In the previous question,
A) the heavier one
which box has the larger
B) the lighter one
velocity after the force acts?
C) both the same
Conceptual Quiz
In the previous question,
A) the heavier one
which box has the larger
B) the lighter one
velocity after the force acts?
C) both the same
The force is related to the acceleration by Newton’s
Second Law (F = ma). The lighter box therefore has
the greater acceleration and will reach a higher
speed after the 1-second time interval.
Follow-up: Which box has gone a larger distance after the force acts?
Follow-up: Which box has gained more KE after the force acts?
Conceptual Quiz
A bowling ball and a Ping-Pong
ball are rolling toward you with
the same momentum. If you exert
the same force to stop each one,
which takes a longer time to bring
to rest?
A) the bowling ball
B) same time for both
C) the Ping-Pong ball
D) impossible to say
p
p
Conceptual Quiz
A bowling ball and a Ping-Pong
ball are rolling toward you with
the same momentum. If you exert
the same force to stop each one,
which takes a longer time to bring
to rest?
We know:
p
Fav 
t
A) the bowling ball
B) same time for both
C) the Ping-Pong ball
D) impossible to say
so p = Fav t
Here, F and p are the same for both balls!
It will take the same amount of time
to stop them.
p
p
Conceptual Quiz
A bowling ball and a Ping-Pong
ball are rolling toward you with the
same momentum. If you exert the
A) the bowling ball
B) same distance for both
same force to stop each one, for
C) the Ping-Pong ball
which is the stopping distance
D) impossible to say
greater?
p
p
Conceptual Quiz
A bowling ball and a Ping-Pong
ball are rolling toward you with the
same momentum. If you exert the
A) the bowling ball
B) same distance for both
same force to stop each one, for
C) the Ping-Pong ball
which is the stopping distance
D) impossible to say
greater?
Use the work-energy theorem: W = KE.
The ball with less mass has the greater
speed (why?), and thus the greater KE (why
again?). In order to remove that KE, work
must be done, where W = Fd. Because the
force is the same in both cases, the
distance needed to stop the less massive
ball must be bigger.
p
p
Conceptual Quiz
Amy (150 lbs) and Gwen (50 lbs) are
standing on slippery ice and push off
each other. If Amy slides at 6 m/s, what
speed does Gwen have?
A) 2 m/s
B) 6 m/s
C) 9 m/s
D) 12 m/s
E) 18 m/s
150 lbs
50 lbs
Conceptual Quiz
Amy (150 lbs) and Gwen (50 lbs) are
standing on slippery ice and push off
each other. If Amy slides at 6 m/s, what
speed does Gwen have?
A) 2 m/s
B) 6 m/s
C) 9 m/s
D) 12 m/s
E) 18 m/s
The initial momentum is zero,
so the momenta of Amy and
Gwen must be equal and
opposite. Because p = mv,
then if Amy has three times
more mass, we see that
Gwen must have three times
more speed.
150 lbs
50 lbs