TEST 1

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TEST 1

• Tuesday May 19 930 amCNH-104Kinematics,
  Dynamics
• Momentum

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Momentum

• Newtons original quantity of motion
• a conserved quantity
• a vector

• Newtons Second Law in another form
• Momentum and Momentum Conservation

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Definition The linear momentum p of a particle
is its mass times its velocity
p ? mv
Momentum is a vector, since velocity is a
vector. Units kg m/s (no special name).
(We say linear momentum to distinguish it from
angular momentum, a different physical quantity.)
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Example 1 2

• A car of mass 1500kg is moving with a velocity of
  72km/h. What is its momentum ?
• Two cars, one of mass 1000kg is moving at 36km/h
  and the other of mass 1200kg is moving at 72km/h
  in the opposite direction. What is the momentum
  of the system ?

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Concept Quiz

• Which object will have the smallest momentum ?
• A 1.6x10-19kg particle moving at 1x103m/s
• A 1000kg car moving at 2m/s
• A 100kg person moving at 10m/s
• A 5000kg truck at rest

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Newtons Second Law
If mass is constant, then the rate of change of
(mv) is equal to m times the rate of change of v.
We can rewrite Newtons Second Law
or
Net external force rate of change of momentum
This is how Newton wrote the Second Law. It
remains true in cases where the mass is not
constant.
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Example 3
Rain is falling vertically into an open railroad
car which moves along a horizontal track at a
constant speed. The engine must exert an extra
force on the car as the water collects in it (the
water is initially stationary, and must be
brought up to the speed of the train).
Calculate this extra force if
v 20 m/s
The water collects in the car at the rate of 6 kg
per minute
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Solution
Plan The momentum of the car increases as it
gains mass (water). Use Newtons second law to
find F.
v is constant, and dm/dt is 6 kg/min or 0.1 kg/s
(change to SI units!), so F (0.1 kg/s) (20
m/s) 2.0 N
F and p are vectors we get the horizontal force
from the rate of increase of the horizontal
component of momentum.
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The total momentum of a system of particles is
the vector sum of the momenta of the individual
particles
ptotal p1 p2 ... m1v1 m2v2 ...
Since we are adding vectors, we can break this up
into components so that
px,Tot p1x p2x .Etc.
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Example 4

• A particle of mass 2kg is moving with a velocity
  of 5m/s and an angle of 45o to the horizontal.
  Determine the components of its momentum.

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Newtons 3rd Law and Momentum Conservation
Two particles interact
Dp1 F12 Dt Dp2 F21 Dt
Newtons 3rd Law F21 -F12
The momentum changes are equal and opposite the
total momentum p p1
p20 doesnt change.
The fine print Only internal forces act.
External forces would transfer momentum into or
out of the system. eg particles moving through
a cloud of gas
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• Conservation of momentum simply says that the
  initial and final momenta are equalpipf
• Since momentum is a vector, we can also express
  it in terms of the components. These are
  independently conserved pixpfx
  piypfy pizpfz

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Concept Quiz

• A subatomic particle may decay into two (or more)
  different particles. If the total momentum
  before is zero before the decay before, what is
  the total after?
• 0 kg m/s
• depends on the masses
• depends on the final velocities
• depends on b) and c) together

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10 min rest
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Collisions

• Conservation of Momentum
• Elastic and inelastic collisions

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Collisions
A collision is a brief interaction between two
(or more) objects. We use the word collision
when the interaction time ?t is short relative to
the rest of the motion. During a collision, the
objects exert equal and opposite forces on each
other. We assume these internal forces are much
larger than any external forces on the
system. We can ignore external forces if we
compare velocities just before and just after the
collision, and if the interaction force is much
larger than any external force.
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Elastic and Inelastic Collisions
Momentum is conserved in collisions. Kinetic
energy is sometimes conserved it depends on the
nature of the interaction force. A collision is
called elastic if the total kinetic energy is the
same before and after the collision. If the
interaction force is conservative, a collision
between particles will be elastic (eg billiard
balls). If kinetic energy is lost (converted to
other forms of energy), the collision is called
inelastic (eg tennis ball and a wall). A
completely inelastic collision is one in which
the two colliding objects stick together after
the collision (eg alien slime and a spaceship).
Kinetic energy is lost in this collision.
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If there are no external forces, then the total
momentum is conserved
p1,i p2,i p1,f p2,f
This is a vector equation. It applies to each
component of p separately.
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Elastic Collisions
In one dimension (all motion along the x-axis)
1) Momentum is conserved
In one dimension, the velocities are represented
by positive or negative numbers to indicate
direction.
2) Kinetic Energy is conserved
We can solve for two variables if the other four
are known.
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One useful result for elastic collisions, the
magnitude of the relative velocity is the same
before and after the collision v1,i v2,i
v1,f v2,f (This is true for elastic
collisions in 2 and 3 dimensions as well).

• An important case is a particle directed at a
  stationary target (v2,i 0)
• Equal masses If m1 m2, then v1,f will be
  zero (1-D).
• If m1 lt m2, then the incident particle recoils
  in the opposite direction.
• If m1 gt m2, then both particles will move
  forward after the collision.

before
after
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Elastic collisions, stationary target (v2,i 0)

• Two limiting cases
• If m1 ltlt m2 , the incident particle rebounds with
  nearly its original speed.

2) If m1 gtgt m2 , the target particle moves
away with (nearly) twice the original speed of
the incident particle.
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Concept Quiz
A tennis ball is placed on top of a basketball
and both are dropped. The basketball hits the
ground at speed v0. What is the maximum speed at
which the tennis ball can bounce upward from the
basketball? (For maximum speed, assume the
basketball is much more massive than the tennis
ball, and both are elastic).

1. v0
2. 2v0
3. 3v0

?
v0
v0
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Example

• An angry 60.0 kg physicist standing on a
  frozen lake throws a 0.5 kg stone to the east
  with a speed of 24.0 m/s.
• Find the recoil velocity of the angry
  physicist.

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Example inelastic collision
A neutron, with mass m 1 amu (atomic mass
unit), travelling at speed v0, strikes a
stationary deuterium nucleus (mass 2 amu), and
sticks to it, forming a nucleus of tritium. What
is the final speed of the tritium nucleus?
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An elastic collision
Two carts moving toward each other collide and
bounce back. If cart 1 bounces back with v2m/s,
what is the final speed of cart 2 ?
5 m/s
6 m/s
4 kg
2 kg
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Example

• A bullet of mass m is shot into a block of mass M
  that is at rest on the edge of a table. If the
  bullet embeds in the block, determine the
  velocity of the bullet if they land a distance of
  x from the base of the table, which has a height
  of h.

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10 min rest
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Impulse
, or dp F dt
Newton 2
For a constant force, Dp F Dt . The vector
quantity F Dt is called the Impulse
J F?t ?p
(change in p) (total impulse from external
forces)
(Newtons Second Law again)
(Extra) In general (force not constant), we
integrate
where the integral gives the area under a curve
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Impulse is the area under the curve. The average
force is the constant force which would give the
same impulse.
The impulse momentum theorem, JF?t?p, tells
us thatwe do not need to know the details of the
force/interaction,we only need to know the area
under the curveintegralof the force.
Compare with work W F Dx so the work-energy
theorem (derived from Newton 2) is DK F Dx.
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Quiz

• A 100g rubber and a 100g clay ball are thrown at
  a wall with equal speed. The rubber ball bounces
  back while the clay ball sticks to the
  wall.Which ball exerts a larger impulse on the
  wall ?
• the clay bass b/c it sticks
• the rubber ball b/c it bounces
• they are equal because the have equal momenta
• neither exerts an impulse b/c the wall doesnt
  move

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Example 1

• A golf ball is launched with a velocity of 44
  m/s. The ball has a mass of 50g. Determine the
  average force on the ball during the collision
  with the club, if the collision lasted 0.01 s.

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Example 2

• Use the impulse-momentum theorem to find how long
  a falling object takes to increase its speed from
  5.5m/s to 10.4m/s.

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Example 3

• A 150 g baseball is thrown with a speed of 20m/s.
  It is hit straight back toward the pitcher at a
  speed of 40m/s. The interaction force is shown
  by the graphWhat is the maximum force
  Fmax that the bat exerts on the ball ?

F
Fmax
t
6ms