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Impulse
For a constant force, impulse is defined as
Ԧ
𝐽Ԧ = 𝐹Δ𝑡
For a force that depends on time
𝐽Ԧ = න 𝐹Ԧ 𝑑𝑡 = 𝐹Ԧ𝐴𝑣𝑔 Δ𝑡
Has units of kg m/s
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Momentum
The momentum of a particle is given by
𝑝Ԧ = 𝑚𝑣Ԧ
Impulse equals the change in momentum
𝐽Ԧ = Δ𝑝Ԧ
Newton’s second law can be written in terms of momentum
𝐹Ԧ =
𝑑𝑝Ԧ
𝑑𝑡
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Conceptual Question
The graph at right shows the
force acting on a particle as a
function of time. If the particle is
initially at rest, at what time is the
velocity of the particle the
highest?
F
2
4
6
t (s)
A. 2 s
B. 4 s
C. 6 s
D. Not enough information
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Example #1
A baseball has a mass of about 145 g. A professional pitcher
can throw a baseball at about 45 m/s. A great hitter can hit a
baseball at about 55 m/s. If a hitter hits a pitch directly back at
the pitcher as hard has he can, find
a) the momentum of the ball before it is hit
b) the momentum of the ball after it is hit
c) the impulse delivered by the bat to the ball
d) If the bat and ball are in contact for 0.7 milliseconds, what
is the average force of the bat on the ball?
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Conservation of Momentum
For a single particle, the momentum is conserved in the
absence of external forces
𝑑 𝑝Ԧ
=0
𝑑𝑡
𝑝Ԧ𝑖 = 𝑝Ԧ𝑓
When two particles collide, the total momentum is conserved
𝑝Ԧ1𝑖 + 𝑝Ԧ2𝑖 = 𝑝Ԧ1𝑓 + 𝑝Ԧ2𝑓
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Collisions in One Dimension
Momentum is conserved in every collision
𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖 = 𝑚1 𝑣1𝑓 + 𝑚2 𝑣2𝑓
Elastic Collisions
Kinetic energy is also conserved
Inelastic Collisions
Kinetic energy is not conserved
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1
1
1
𝑚 𝑣2 + 𝑚 𝑣2 = 𝑚 𝑣2 + 𝑚 𝑣2
2 1 1𝑖 2 2 2𝑖 2 1 1𝑓 2 2 2𝑓
Completely Inelastic Collisions
𝑣1𝑓 + 𝑣1𝑖 = 𝑣2𝑓 + 𝑣2𝑖
In a perfectly inelastic collision, two masses
stick together
𝑣1𝑓 = 𝑣2𝑓 =
𝑚1 𝑣1𝑖 + 𝑚2 𝑣2𝑖
𝑚1 + 𝑚2
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Conceptual Question
An innocent mosquito is flying along the interstate
and has a collision with a speeding semi. Which
one feels a greater force?
A) The mosquito
B) The semi
C) They both feel the same force
D) There are no forces involved
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Conceptual Question
An innocent mosquito is flying along the interstate
and has a collision with a speeding semi. Which
one experiences a greater change in
momentum?
A) The mosquito
B) The semi
C) They both experience the same change in
momentum
D) Neither one experiences a change in
momentum
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Conceptual Question
An innocent mosquito is flying along the interstate
and has a collision with a speeding semi. Which
one experiences a greater acceleration (change
in velocity)?
A) The mosquito
B) The semi
C) They both experience the same acceleration
D) Neither one experiences an acceleration
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Example #2
v0
A piece of putty with horizontal velocity v0 = 20 m/s and mass
mp = 30g impacts a block with mb = 270g which is initially at rest
on a frictionless surface.
a) If the putty sticks to the block, what is the final velocity of
the system?
b) What is the energy lost in the collision?
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Example #3
A tennis ball with a small mass m is
released on top of a basketball with
a large mass M. After the basketball
hits the ground, it has a velocity v0 =
5 m/s pointing straight up and the
tennis ball has a velocity v0 = 5 m/s
pointing straight down. After the
tennis ball and the basketball
collide, what is the velocity of each?
(Assume that M >> m)
v0
v0
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Collisions in Two Dimensions
𝑝Ԧ1𝑖 + 𝑝Ԧ2𝑖 = 𝑝Ԧ1𝑓 + 𝑝Ԧ2𝑓
In two dimensions, this can be broken up into vector components
𝑝1𝑥𝑖 + 𝑝2𝑥𝑖 = 𝑝1𝑥𝑓 + 𝑝2𝑥𝑓
𝑝1𝑦𝑖 + 𝑝2𝑦𝑖 = 𝑝1𝑦𝑓 + 𝑝2𝑦𝑓
In an elastic collision, the total kinetic energy is conserved
1
1
2
2
2
2
𝑚1 (𝑣1𝑥𝑖
+ 𝑣1𝑦𝑖
) + 𝑚2 (𝑣2𝑥𝑖
+ 𝑣2𝑦𝑖
)
2
2
1
1
2
2
2
2
= 𝑚1 (𝑣1𝑥𝑓
+ 𝑣1𝑦𝑓
) + 𝑚2 (𝑣2𝑥𝑓
+ 𝑣2𝑦𝑓
)
2
2
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Conceptual Question
Two cyclists with roughly equal mass and equal
velocity collide at a right angle and stick
together in an intersection. Which arrow best
represents the direction that they move after
the collision?
A
B
C
D
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Example #4
A car with mass 1500 kg traveling
east at 25 m/s collides at an
intersection with a 2500kg van
traveling north at a speed of 20
m/s. Find the magnitude and
direction of the velocity of the
wreckage after the collision,
assuming that the vehicles undergo
a perfectly inelastic collision and
assuming no friction.
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Conceptual Question
A cue ball strikes another billiard ball which is initially at
rest. The initial and final velocities of the cue ball are shown
below. Which arrow best represents the final velocity of the
eight ball?
𝑣Ԧ𝑓
𝑣Ԧ0
A
B
C
D
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Example #5
A proton collides elastically with another proton that is
initially at rest. The incoming proton has an initial speed of
3.5 × 105 𝑚/𝑠, and they collide as shown in the figure
(glancing collision). After the collision, one proton has a final
speed of 2.8 × 105 𝑚/𝑠. Find the final velocity (magnitude
and direction) of each proton.
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Center of Mass
In one dimension, the center of mass for a system of particles
is given by
𝑀𝑥𝐶𝑀 = ෍ 𝑚𝑖 𝑥𝑖 = 𝑚1𝑥1 + 𝑚2 𝑥2 + 𝑚3 𝑥3 + ⋯
𝑖
𝑀 = ෍ 𝑚𝑖 = 𝑚1 + 𝑚2 + 𝑚3 + ⋯
𝑖
𝑥𝐶𝑀 =
1
෍ 𝑚𝑖 𝑥𝑖
𝑀
𝑖
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Center of Mass
In one dimension, the center of mass for a pair of masses is
given by
𝑚1 𝑥1 + 𝑚2 𝑥2
𝑥𝐶𝑀 =
𝑚1 + 𝑚2
And the velocity of the center of mass is given by
𝑣𝐶𝑀 =
𝑚1𝑣1 + 𝑚2 𝑣2
𝑚1 + 𝑚2
If there are no external forces, then 𝑣𝐶𝑀 is conserved.
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Center of Mass
For a system of particles, we can define the center of mass
𝑀𝑟Ԧ𝐶𝑀 = ෍ 𝑚𝑖 𝑟Ԧ𝑖
න 𝑟𝑑𝑚
Ԧ
𝑖
𝑀𝑣Ԧ𝐶𝑀 = ෍ 𝑚𝑖 𝑣Ԧ𝑖
න 𝑣𝑑𝑚
Ԧ
𝑖
𝑀 𝑎Ԧ𝐶𝑀 = ෍ 𝑚𝑖 𝑎Ԧ 𝑖
න 𝑎𝑑𝑚
Ԧ
𝑖
Newton’s second law for the system of particles can then be
written as
𝐹Ԧ𝑁𝐸𝑇 = 𝑀 𝑎Ԧ 𝐶𝑀
𝐹Ԧ𝑁𝐸𝑇 = ෍ 𝐹Ԧ𝑒𝑥𝑡,𝑖
𝑖
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Example #6
Four masses are connected by
massless rods in the shape of a square
with side length L = 1 m, as shown in
the diagram at right. What is the
position of the center of mass of this
system?
2 kg
4 kg
3 kg
1 kg
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Example #7
A Chihuahua walks from one end of a skateboard to the other. The
skateboard has a mass of 2 kg and a length of 75 cm, and the
Chihuahua has a mass of 3 kg. If the skateboard can roll without
friction, how far does the skateboard move?
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