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October 27, 2006 2
Momentum and the 2nd Law
Momentum and the 2nd Law
"momentuDefinition: m" p mv
; ; ;x x y y z zp mv p mv p mv
nd ( ) 2 Law:
dv d mv dpF ma m
dt dt dt
2
1
2 1 ( )t
x x x x
t
p p p F t dt
Momentum is a vector quantity.
Force changes momentum.
Momentum Units: kg m/sor N s.
October 27, 2006 3
Solving Momentum Problems
Solving Momentum Problems
Drawing a before-and-after pictorial representation
Sketch the situation. Use two drawings, labeled “Before” and “After,” to show the objects before they interact and again after they interact.
Establish a coordinate system. Select your axes to match the motion. Define symbols. Define symbols for the masses and for the velocities
before and after the interaction. Position and time are not needed. List known information. Give the values of quantities known from the
problem statement or that can be found quickly with simple geometry or unit conversions. Before-and-after pictures are usually simpler than the pictures you used for dynamics problems, so listing known information on the sketch is adequate.
Identify the desired unknowns. What quantity or quantities will allow you to answer the question? These should have been defined as symbols in Step 3 above.
October 27, 2006 4
Colliding Train CarsColliding Train Cars
1 1 2 2 1 1 2 2( ) ( ) ( ) ( )fx fx ix ixm v m v m v m v
1 2 2 0f f f i im v m v mv mv mv
1
2f iv v
Cars coupleCar 2 at rest m1=m2=m
October 27, 2006 5
A System of N ParticlesA System of N ParticlesThe N=3 Case: For every pair of particles, the action/reaction pairs Fj on k and Fk on j are equal and opposite force vectors. In addition, each particle may be subjected to possible external forces Fext on k from agents outside the system.
Defining “The System”: In considering the conservation of momentum, we will normally define “the system” as a group of interacting particles NOT subject to external forces.
October 27, 2006 6
The Law ofConservation of Momentum
The Law ofConservation of Momentum
ext on k net1
Conclusion: N
k
dPF F
dt
netIf 0 then 0
So constant Conservation of Momentum
dPF
dt
P
1 1
N Nk
kk k
d pdPF
dt dt
1 2 31
Consider a system of N interacting
particles. The total momentum of
the system is:
...N
N kk
P p p p p p
Law of Conservation of Momentum: The total momentum P of an isolated system is a constant. Interactions within the system do not change the system’s total momentum.
October 27, 2006 7
Example:Two Balls Shot from a Tube
Example:Two Balls Shot from a Tube
1 1 2 2 1 1 2 2( ) ( ) ( ) ( ) 0fx fx ix ixm v m v m v m v
121 2 23
1
( ) ( ) ( ) 2.0 m/sfx fx fx
mv v v
m
A 10 g ball and a 30 g ball are placed in a tube with a massless spring between them. When the spring is released, the 10 gm ball flies out of the tube at a speed of 6.0 m/s.
With what speed does the 30 g ball emerge from the other end?
October 27, 2006 8
MODEL: Clearly define the system• If possible, choose a system that is isolated or within which the interactions are sufficiently short and intense that you can ignore external forces for the duration of the interaction (the impulse approximation). Momentum is conserved.• If it’s not possible to choose an isolated system, try to divide the problem into parts such that momentum is conserved during one segment of the motion. Other segments of the motion can be analyzed using Newton’s laws or, as you’ll learn in Chapters 10 and 11, conservation of energy.
VISUALIZE: Draw a before-and-after pictorial representation. Define symbols that will be used in the problem, list known values, and identify what you’re trying to find.SOLVE: The mathematical representation is based on the law of conservation of momentum:
Strategy:Conservation of Momentum
Strategy:Conservation of Momentum
net 0F
f iP P
*
In component form, this is: f f f i i i1 2 3 1 2 3x x x x x xp p p p p p
fy fy fy iy iy iy1 2 3 1 2 3p p p p p p
ASSESS: Check that your result has the correct units, is reasonable, and answers the question.
October 27, 2006 9
Example: Rolling AwayExample: Rolling Away
Bob sees a stationary cart 8.0 m in front of him. He decides to run to the cart as fast as he can, jump on, and roll down the street. Bob has a mass of 75 kg and the cart’s mass is 25 kg. If Bob accelerates at a steady 1.0 m/s2 before jumping on, what is the cart’s speed just after Bob jumps on?
2 21 0 1 0 1( ) ( ) 2 ( ) 2x B x B x xv v a x x a x
21 1( ) 2 2(1.0 m/s )(8.0 m) 4.0 m/sx B xv a x
2 2 2
1 1
1
( ) ( ) ( )
( ) ( )
( )
B x B C x C B C x
B x B C x C
B x B
m v m v m m v
m v m v
m v
2 1( )
(75 kg)(4.0 m/s) 3.0 m/s
(75 kg) (25 kg)
Bx x B
B C
mv v
m m
October 27, 2006 10
It Depends on the System
It Depends on the SystemIs momentum conserved in the system?
No!(external force)
Yes!(mostly, neglect Sun’s force?)
Goal: To chose a system in which momentum IS conserved.
