Transcript 16. Impulse

Momentum
The secret of collisions and
explosions
Presentation ©2003 Dr. Phil Dauber
as modified by R. McDermott
Is it Harder to Stop..
A small car moving
at 20 mph?
Or a truck moving at
1 mph?
Is it mass or velocity that matters?
Mass or Velocity?
Mass and velocity are both important!
It’s difficult to tell which vehicle will be
harder to stop without knowing the
mass and the velocity for both objects
A small object that is moving fast may
be just as hard to stop as a larger
object moving slowly
Who Pushes Who?
Rin Tin Tin and the Refrigerator meet at
the 50 yard line
Mass 20 Kg
Mass 160 Kg
Speed 17 m/s
Speed 2 m/s
Who pushes who over the 50 yard line?
Momentum = Mass x Velocity
Whoever has the most momentum = mv
wins!
Mass 20 Kg
Mass 160 Kg
Speed 17 m/s
Speed 2 m/s
Momentum = 340 Kg m/s
Momentum = 320 Kg m/s
Momentum is a Vector
p = mv
Force is required to change the momentum of
an object. Newton stated his 2nd law:
SF = Dp/Dt
The rate of change of momentum of a body
equals the net force exerted on it (equivalent
to F = ma)
Proof of Equivalence of the
Forms of Newton’s 2nd Law
SF = Dp/Dt = (mv –mv0)/Dt = m(v - v0)/ Dt
SF = mDv/Dt = ma
Q: Which form of the law is more
general? (which includes the
possibility that the mass could
change?)
Example:
Water leaves a hose at a rate of 1.5
kg/s with speed 20 m/s and hits a car
without splashing back. What force
is exerted by the water on the car?
F = Dp/Dt = (pfinal - pinitial )/Dt
F = (0 – 30kg-m/s)/1s = -30N
Momentum is Conserved
The total momentum of an
isolated system of bodies remains
the same
“isolated” means net external
force is zero
Momentum before = momentum
after
m1v1 + m2v2 = m1v1’ + m2v2’
Applies to collisions and explosion
events
Conservation #1:
Railroad Cars Collide
A 10,000 kg railcar moving 24 m/s hits and
locks with an identical car at rest. What is
the final speed of the two cars?
m1v1 + m2v2 = m1v1’+ m2v2’
(10,000 kg) (24 m/s) + 0 = 2.4 x 105 kg-m/s = (m1 + m2)v’
v’= (2.4 x 105 kg-m/s)/2.0 x 105 kg = 12 m/s
Example #2
Conservation #2:
Example #3
Example #4
Example #5
Rebound
Bertha, the bouncing ewe, hits the
ground at 5.0 m/s and rebounds at
the same speed. If her mass is 20kg,
and she is in contact with the ground
for 0.1 seconds, what force does she
exert on the ground?
Answer:
2,000 N
Recoil of a pistol
What is the recoil velocity of a 1 kg pistol
that shoots a .02 kg bullet at 400 m/s?
Initial momentum = 0 = mBvB+mpvp
0 = (0.02kg x 400 m/s) + (1kg)vp
vp = - 20 m/s Q: Does the shooter recoil
too?
Explain This:
As a rock falls to earth, it gains velocity,
so it also gains momentum. Is
momentum conserved in this case?
Impulse
Impulse (J) = FDt = Dp
Impulse is the product of the force and
the time during which the force acts
Impulse equals change in momentum
F is usually non-uniform and time
interval is usually short
What is Impulse For?
The impulse form of Newton’s
2nd Law is especially useful when
the time interval for the applied
force can be varied
This situation is often found in
athletic activities, and is of
fundamental concern in the
engineering of manned vehicles.
Why Should You…
Bend your knees when you land?
Pull back when the baseball
enters your mitt?
Follow through when you swing?
Not walk into a punch? (like
Mike Tyson did)
Why do…
Cars have crumple zones?
Sprinters take short steps?
Martial artists hit through
their target?
Bullets striking a bullet-proof
vest sometimes break ribs?
Elastic and Inelastic Collisions
Elastic – kinetic energy is conserved as well
as momentum and total energy
Inelastic – kinetic energy is not conserved –
some energy turns into heat
Elastic – bounce
Completely inelastic - stick
How Can You Tell?
Any collision that produces deformation
is inelastic, even if the objects return to
their original shapes
Any collision in which the objects stick
together is inelastic
Any collision that produces sound or
heat is inelastic
In real life, elastic collisions only occur
with atom-sized objects!
The Situation:
The moving mass 1 collides elastically
with mass 2 at rest
The masses are equal
The collision is head-on
Elastic Collision
Final State:
Initial State:
m1
m2 = m1
Same
masses,
Unknown v1’
and v2’
v1
v2 = 0
Observations
What actually happens in such a
collision?
The moving mass stops and the
stopped one moves off at the same
speed as the incoming one
There appears to be a total transfer of
momentum
The Physics
Conservation of Momentum
Total the same before and after
Conservation of Kinetic Energy
Total the same before and after
The Math - Momentum
m1v1 +m2v2 = m1v1’ + m2v2’
Let:
v1 = v and v2 = 0
Let:
m1 = m2 = m
Then:
mv = mv1’ + mv2’
v1’ = v – v2’
And:
Kinetic Energy Conservation
KEinitial = Kefinal
½ mv2 = ½ mv1’2 + ½ mv2’2
Divide by ½m:
v2 = v1’2 + v2’2
Substitute:
v1’ = v – v2’
Yields:
v2 = (v – v2’)2 + v2’2
Energy cont.
Expanding: v2 =
v2 –2vv2’+v2’2 + v2’2
Simplify:
2vv2’ = 2v22
Divide by 2v2:
v = v2’ or v2’ = v
And also:
v1’ = v – v2’ = 0
QED (It is demonstrated)
v2’ = v Final speed of object
initially struck equals that of object
which struck it.
v1’ = v – v2’ = 0
Final speed of
initially moving object is zero.
Ballistic Pendulum
A bullet of mass m is fired into a block
of wood of mass M suspended from a
string. The bullet remains in the block
which rises a height h. What was the
speed of the bullet? Show that:
v = (2gDh)1/2(m + M)/m
Dh
Collisions in Two Dimensions
Remember momentum p is a vector
x and y components are conserved
separately
q1
q2
What is the total vertical momentum?
2-D Collision:
Acknowledgements
Some diagrams and animations
courtesy of Tom Henderson, Glenbrook
South High School, Illinois