Impulse and Momentum Linear Momentum

5/8/16
Impulse and Momentum
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Linear Momentum
“The change of motion is ever proportional to the motive
force impressed; and is made in the direction of the right
[straight] line in which that force is impressed”
Sir Isaac Newton
What Newton called “motion”
translates into “moving inertia”.
Today the concept of moving inertia
is called momentum which is defined
as the product of mass and velocity.
momentum = mass × velocity
momentum is a
vector quantity
!
!
p = mv
LARGE MASS, SMALL VELOCITY
SI unit for
momentum
1 kilogram × 1
meter
kg ⋅m
=1
second
s
SMALL MASS, LARGE VELOCITY
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Impulse and Momentum
Newton’s 2nd Law was written in terms of momentum and force
F = ma
F=
mΔv
t
F=
Δp
t
IMPULSE MOMENTUM THEOREM
Ft = mΔv
impulse
momentum
change
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Impulse causes a change of momentum for any object. This is
analogous to work, which causes a change of energy for any object.
impulse is a
SI unit of impulse
vector quantity
! !
I = Ft
1 newton × 1 second = 1 N ⋅ s
1 N ⋅s = 1
kg ⋅ m
s
Impulse and Safety
MOMENTUM CHANGED BY A
SMALL FORCE OVER A LONG TIME
Other examples of car safety that
involve increased time and decreased
force (but result in equal impulse)
Airbags
Seatbelts
Crumple zones
Bumpers
Padding
MOMENTUM CHANGED BY A
LARGE FORCE OVER A SHORT TIME
Other examples force/time in impulse
Air cushioned shoes
Bending knees when landing
Natural turf vs. artificial turf
Gym and playground mats
Helmets/padding for sports
Shocks/forks in bicycles
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5/8/16
Impulse of Sports
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A boxer who “rolls with the punch” will experience less force over more time.
IN SPORTS THE
IMPACT TIME IS
SHORT, BUT
EVERY BIT
COUNTS!
In many sports you are taught to “follow through”. Why?
As you “follow through” the time of contact with the ball is
increased, so the amount of momentum change is also increased.
You get more momentum and more speed and more distance!
Impulse of Sports
Ball
Mass
(kg)
speed
imparted
(m/s)
impact
time
(ms)
Baseball
0.149
39
1.25
Football (punt)
0.415
28
8
Golf ball (drive)
0.047
69
1
Handball (serve)
0.061
23
12.5
Soccer ball (kick)
0.425
26
8
Tennis ball (server)
0.058
51
4
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Third Law and Impulses
Every action has an equal
and opposite reaction:
F1 = −F2
Every action takes just as
long as the reaction so:
t1 = t 2
Every impulse has an equal
and opposite impulse:
F1t1 = −F2t 2
The momentum changes are equal and opposite:
m1 Δv1 = −m 2 Δv2
The momentum changes are equal and opposite:
Δp1 = −Δp2
or
Δp1 + Δp2 = 0
Conservation of Momentum
If two isolated objects interact (collide or separate), then
the total momentum of the system is conserved (constant).
pi = p f
click for
applet
SEPARATION OR
EXPLOSION
or m1v1i + m2 v2i = m1v1 f + m2 v2 f
click for
applet
INELASTIC COLLISION
click for
applet
ELASTIC COLLISION
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Inelastic Collisions
Most collision are somewhat inelastic, with some
kinetic energy converted into thermal energy
When two objects collide they may combine
into one object in a perfectly inelastic collision
m1v1i + m2 v2i = (m1 + m2 )v f
BALLISTIC PENDULUM
Elastic Collisions
Ideal collisions are elastic, with zero kinetic
energy converted into thermal energy (heat)
Springs and magnets can create elastic
collisions. Other examples: air molecules,
billiard balls, spacecraft gravitational boost.
NEWTON’S CRADLE
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Collision in Two Dimensions (Honors)
m1v1xi + m2 v2 xi = m1v1xf + m2 v2 xf
m1v1yi + m2 v2 yi = m1v1yf + m2 v2 yf
CONSERVATION IN X,Y DIRECTIONS
Two billiard balls, of equal mass, collide as shown below. Ball 1 is initially
traveling along the x-axis with a speed of 15 m/s, and ball 2 is at rest. After the
collision, ball 1 moves away with a speed of 6.6 m/s at an angle of 58˚. What is the
speed of ball 2 after the collision? What is the angle made with the positive x axis?
Is the collision elastic?
mv1xi = mv1xf + mv2 xf 15 = 6.6 cos 58˚+v2 xf
mv1yi = mv1yf + mv2 yf 0 = 6.6sin 58˚+v2 yf
v2 xf = 11.5 m/s
v2 yf = −5.60 m/s
v2 f = 11.5 2 + (−5.60)2 = 7.05 m/s
⎛ −11.5 ⎞
θ = tan −1 ⎜
= −25.9˚
⎝ 4.29 ⎟⎠
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