Linear Momentum
Linear Momentum
Momentum is a measure of how hard it is
to stop or turn a moving object.
 p = mv (single particle)
 P = Σpi (system of particles)

• Problem: How fast must an electron move to have the same
momentum as a proton moving at 300 m/s?
• Problem: A 90-kg tackle runs north at 5.0 m/s and a 75-kg
quarterback runs east at 8.0 m/s. What is the momentum of the
system of football players?
Impulse (J)
Impulse is the integral of force over a period
of time.
 J =  Fdt
 The change in momentum of a particle is
equal to the impulse acting on it.
 Dp = J


Problem: Restate Newton’s 2nd Law in terms of impulse.
Force
Impulsive forces are generally of
high magnitude and short duration.
time
Problem: A 150-g baseball moving at 40 m/s 15o below the horizontal is
struck by a bat. It leaves the bat at 55 m/s 35o above the horizontal.
What is the impulse exerted by the bat on the ball?
If the collision took 2.3 ms, what was the average force?
• Problem: An 85-kg lumberjack stands at one end of a floating 400-kg
log that is at rest relative to the shore of a lake. If the lumberjack jogs to
the other end of the log at 2.5 m/s relative to the shore, what happens to
the log while he is moving?
• Problem: Two blocks of mass 0.5 kg and 1.5 kg are placed on a
horizontal, frictionless surface. A light spring is compressed between
them. A cord initially holding the blocks together is burned; after this, the
block of mass 1.5 kg moves to the right with a speed of 2.0 m/s. A) What
is the speed and direction of the other block? B) What was the original
elastic energy in the spring?
Conservation of Linear Momentum

The linear momentum of a system is
conserved unless the system experiences
an external force.
Collision Review
Collisions in 1 and 2
Dimensions
Collision review


In all collisions, momentum is conserved.
Elastic Collisions: No deformation occurs
 Kinetic

Inelastic Collisions: Deformation occurs
 Kinetic

energy is lost.
Perfectly Inelastic Collisions
 Objects

energy is also conserved.
stick together, kinetic energy is lost.
Explosions
 Reverse
of perfectly inelastic collision, kinetic
energy is gained.
• 1 Dimensional Problem: A 1.5 kg cart traveling at 1.5 m/s
collides with a stationary 0.5 kg cart and sticks to it. At
what speed are the carts moving after the collision?
• 1 Dimensional Problem: A 1.5 kg cart traveling at 1.5 m/s
collides elastically with a stationary 0.5 kg cart. At what
speed are each of the carts moving after the collision?
• 1 Dimensional Problem: What is the recoil velocity of a 120kg cannon that fires a 30-kg cannonball at 320 m/s?
For 2-dimensional collisions
Use conservation of momentum
independently for x and y dimensions.
 You must resolve your momentum vectors
into x and y components when working the
problem

•2 Dimensional Problem: A pool player hits a cue ball in the x-direction at 0.80
m/s. The cue ball knocks into the 8-ball, which moves at a speed of 0.30 m/s at
an angle of 35o angle above the x-axis. Determine the angle of deflection of the
cue ball.
Collision Problems
Practice Day
Problem 1

A proton, moving with a velocity of vii, collides elastically with another
proton initially at rest. If the two protons have equal speeds after the
collision, find a) the speed of each in terms of vi and b) the direction of the
velocity of each.
Center of Mass
Center of Mass
The center of mass of a system is the
point at which all the mass can be
assumed to reside.
 Sometimes the system is an assortment of
particles and sometimes it is a solid object.
 Mathematically, you can think of the center
of mass as a “weighted average”.

Center of Mass – system of points
xi mi

 Analogous
xcm 
M
equations exist for
velocity
and
y
m

i i
ycm 
acceleration, for
M
example
zi mi

zcm 
vx ,i mi

M
vx ,cm 
M

Sample Problem
y
2
Determine the
Center of Mass.
2 kg
x
0
3 kg
1 kg
-2
-2
0
2
4
Center of Mass
-- solid objects
x

If the object is of uniform
density, you pick the
geometric center of the
object.
x
x
x

x
For combination objects,
pick the center of each
part and then treat each
center as a point.
Center of Mass Problem
y

2R
x
Determine the
center of mass
of this shape.
Center of Mass for more
complicated situations

If the shapes are points or simple geometric
shapes of constant density, then
xcm

xm


i
i
M
Anything else is more complicated, and
xcm
xdm


M
• Problem: Find the x-coordinate of the center of mass of a rod of length L
whose mass per unit length varies according to the expression l = ax.
y
L
x
• Problem: A thin strip of material of mass M is bent into a
semicircle of radius R. Find its center of mass.
y
R
-R
R
x
Motion of a System
of Particles
Motion of a System of Particles



If you have a problem involving a system of
many particles, you can often simplify your
problem greatly by just considering the motion of
the center of mass.
If all forces in the sytem are internal, the motion
of the center of mass will not change.
The effect of an external force that acts on all
particles can be simplified by considering that it
acts on the center of mass; that is:
SFext = Macm
Homework problem #47 – collaborative work
(15 minutes)


This problem is a toughie! However, it gets easier if you
use the idea that the center of mass of the system of
Romeo, Juliet, and canoe does not change because all
forces are internal. The question is paraphrased below:
Rome (77 kg) sits in the back of a canoe 2.70 m away
from Juliet (55 kg) who sits in the front. The canoe has a
mass of 80 kg. Juliet moves to the rear of the canoe to
kiss Romeo. How far does the canoe move forward
when she does this? (Assume a canoe that is
symmetrical).