Recap:

1
Conservation of Momentum Collisions (Chapter 7)

• Recap
• For situations involving an impact or a
  collision, where large forces exists for a very
  small time we define
• Impulse F x ?t (units N.s) - a
  vector
• where F is the force and ?t is the time
  of action.
• By Newtons 2nd law (F m.a) we determined
• Impulse F. ?t m. ?v
• or Impulse Change in momentum (?P)
• Result Large impulses cause large changes in
  motion!

impulse (F. ?t)
momentum change
m
(?P m. ?v)
2
Momentum

• Momentum (P) is the product of the mass of an
  object and its velocity
• P m.v (units kg.m/s)
• Momentum is a vector acting in same direction as
  velocity vector.
• Example 1 A 100kg boulder rolling towards a
  castle gate at 3m/s.
• Momentum boulder P m.v 100 x 3 300
  kg.m/s
• Example 2 A 1 kg missile flying towards the
  castle gate at 300 m/s (speed of sound).
• Momentum missile P m.v 1 x 300 300
  kg.m/s
• Result Different objects can have the same
  momentum (quantity of motion)
• But its the change in momentum of an object that
  is important as this equals the impulse
• ?P m. ?v impulse

3
Impulse - Momentum Principle

• An impulse acting on an object causes a change
  in its momentum.
• The change in momentum is equal in magnitude and
  direction to the applied impulse.
• Impulse m. ?v
• A new way of looking at Newtons 2nd law!
• Example A 50kg rock is hurled by a giant
  catapult with a force of 400 N applied for 0.5
  sec.
• Impulse F. ?t 4000 x 0.5 2000 N.s
• Thus Change in momentum 2000 Ns m. ?v
• or ?v 40 m/s ( 140 km/hr)
• Note When the initial velocity is zero
• change in momentum objects momentum
• change in velocity objects velocity

2000 50
4

• Ex Balls Change in Momentum

1. Assume no energy is lost, therefore KE of the
   ball is the same before and after impact. (KE
   ½m.v2)
2. On impact the momentum of ball is decreased to
   zero.

?P
P2m.v
P1m.v
impulse
before during after

• 3. The total change in momentum ?P is
• ?P P2 P1 m.v (- m.v)
• ?P 2 m.v
• The impulse required to change the direction of
  ball is therefore equal to twice momentum of the
  impacting ball.
• (ie. Twice as large as what is needed to
  simply stop the ball.)

(As P1 opposite to P2)
5

• Summary
• Rewriting Newtons 2nd law shows that
• Impulse Change in momentum (?P)
• Impulse F. ?t m. ?v
• Large impulses produce large changes in momentum
  -resulting in large velocity changes.
• Where
• Impulse F x ?t (units N.s) - a vector
• Momentum P m.v (units kg.m/s) - a
  vector.

6
Conservation of Momentum

• A new principle for studying collisions which
  results from Newtons 3rd law applied to the
  impulse/ momentum equation.
• Conservation of momentum enables us to understand
  collisions and to predict many results without a
  detailed knowledge of time varying forces.
• Consider Two sticky objects moving towards each
  otherthey meet in mid-air and after colliding
  stick together and move as one body.
• During the moment of impact there is a strong
  force acting for a time ?t.
• By Newtons 3rd law, an equal and opposite force
  F acts back (remember forces occur in pairs).

7
Conservation of Momentum

• As ?t is the same for both forces, the impulses
  they produce ( F.?t) are the same
  magnitude but in opposite directions.
• By Newtons 2nd law

v2
v1
- impulse
impulse
Impulse Change in momentum (?P m.
?v) - Thus the change in momentum experienced by
both objects must be the samebut in opposite
directions. - The total change in momentum of the
system (i.e. both objects combined) is therefore
ZERO! - In other words , the total momentum of
the system is conserved (i.e. changes of momentum
within system cancel each other out).
8

• Conservation of Momentum
• The total momentum of the system is conserved
  (if no other external forces acting).
• However, different parts of the system can
  exchange momentum (but the total remains the
  same).
• Note If a net external force acts on system,
  then it will accelerate and its momentum will
  change.
• Conservation of momentum allows us to examine
  interesting impact situations

9

• Example Custard pie fights!
• vc -10 m/s vt 0.5
  m/s
• Pc mc.vc Pt mt.vt
• Pc -1 x 10 -10 kg.m/s Pt 100 x 0.5 50
  kg.m/s
• (negative sign as opposite direction)
• Total momentum Ptarget Pcustard pie 50 -
  10 40 kg.m/s
• Question What is the velocity of target and pie
  after impact?
• Total mass mt mc 101 kg
• Total momentum 40 kg.m/s (P m.v)
• v P / m 40 / 101 0.4 m/s
• Result The unsuspecting target has a larger
  initial momentum, so his direction of motion
  prevails but the pie reduces his forward velocity
  (briefly)!

mc 1 kg
mt 100 kg
positive direction
10
Recoil

• A special case of conservation of momentum when
  the initial velocity of the interacting bodies is
  often zero.
• E.g. - two ice skaters pushing off -
  firing a gun
• - rocket propulsion
• We have already looked at what happens to the ice
  skaters motion (using Newtons 3rd law), but now
  we can use conservation of momentum to determine
  their velocities
• Example Initial momentum 0
• Thus total momentum after push off 0
• The momentum of each person must therefore
  be equal but opposite in direction and P2 -P1.
• But, as P1 m1.v1 and P2 m2.v2, the
  velocities will be in opposite directions and
  will depend on their masses.
• Eg. If m1 3m2 then v2 will be 3v1 in opposite
  direction!

11
Firing a Gun (initial momentum zero)

• Momentum of bullet Momentum of gun
• m1v1 - m2v2
• Mass of bullet is small but its velocity is
  highcreating a large recoil.
• To reduce velocity of recoil (v2), hold gun with
  locked arms so the mass m2 becomes mass of gun
  your body.
• Similarly a very massive cannon will jump back
  much less than a light onefor the same shot.
• Rocket propulsion
• Exhaust gasses have large momentum (light
  molecules but very high velocity).
• Momentum gained by rocket in forward direction
  equals momentum of exhaust gasses in opposite
  direction.
• This is why rockets (i.e. recoil) work in outer
  space as gasses and rocket push against each
  other as gasses expelled.

12
Collisions

• Two main types Elastic and Inelastic
• Different kinds of collisions produce different
  resultse.g. sometimes objects stick together and
  other times they bounce apart!
• Key to studying collisions is conservation of
  momentum and energy considerations
• Questions
• - What happens to energy during a collision?
• - Is energy conserved as well as momentum?

13
Perfectly Inelastic Collisions (Sticky ones!)

• E.g. Two objects collide head on and stick
  together, moving as one after collision (only one
  final momentum / velocity to compute).
• Ignoring external forces (which are often low
  compared with large impact forces), we use
  conservation of momentum.
• E.g. Coupling train trucks (low rolling
  friction)
• Before System momentum mv 2x104x10 2x105
  kg.m/s
• After Final system mass (2010515)x103 kg
  50x103 kg
• As final momentum initial momentum
• vfinal
• Thus total momentum of system has remained
  constant but the colliding trucks velocity has
  reduced (i.e. momentum shared).

v 10 m/s
3 stationary trucks
Pfinal total mass
2x105 5x104
4 m/s

14

• Question What happens to the energy of this
  system?
• Total energy Kinetic Energy ½.m.v2 (i.e. no
  PE change)
• Before impact KEtot KEtruck KE3trucks
• ½(2x104)(10)2 0
• 106 Joules (1MJ)
• After impact KEtot ½.mtot.v2tot
  ½(5x104)(4)2
• 2x105 J
• Energy differences (10 - 2)x105 J 8x105 J
  (i.e. 80 loss)
• Results Energy is lost in an inelastic collision
  (heat, sound) and the greatest portion of energy
  is lost in a perfectly inelastic collision when
  objects stick together!
• Extreme example
• Pie (or bullet) hitting a wall All KE is lost on
  impact!

15
Bouncing Collisions

• If objects bounce off one another rather than
  sticking together, less energy is lost in the
  collision.
• Bouncing objects are called either elastic or
  partially inelastic. The distinction is based
  on energy.
• Elastic Collisions
• No energy is lost in an elastic collision.
• E.g. A ball bouncing off a wall / floor with
  no change in its speed (only direction).
• Partially Inelastic Collisions
• In general most collisions are partially
  inelastic and involve some loss of energy as
  they bounce apart.
• Playing pool
• Very little energy is lost when balls hit each
  other and the collision is essentially elastic.
  In such cases
• Momentum and energy are
  conserved.

16

• In an elastic collision we need to find the final
  velocity of both colliding objects.
• Use conservation of momentum and conservation of
  energy considerations
• Example
• Answer The cue ball stops dead on impact and
  red ball moves forward with the same velocity
  (magnitude and direction) as that of the cue ball
  prior to impact!
• Why?...Because both KE( ½.m.v2) and momentum
  (m.v) are conserved on impact.
• As the masses of both balls are the same the only
  solution to conserve both KE and momentum is for
  all the energy and momentum to be transferred to
  the other (red) ball.
• Its a facttry it for yourself!!!

Question What happens to red ball and cue ball?
P1
cue ball
(no spin)
P2
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(No Transcript)
18
Impulse

• Impulse is the average force acting on an object
  multiplied by its time interval of action.
• Impulse F . ?t (units N.s)
• Note Since instantaneous force may vary during
  impact we must use average force.
• Impulse is a vector acting in the direction of
  average force
• The larger the force (F) and the longer it acts
  (?t) the larger the impulse. ( Impulse is
  therefore a measure of the overall effect of the
  force.)
• However Impulse F. ?t m. ?v
• So an impulse causes a change in velocity (?v) in
  magnitude and direction.
• In Newtons words the product m.?v is the
  change in the quantity of motion.
• We now term this the change in momentum.

19

• Other forms of transport
• - Aircraft move by pushing against air.
• - Boats push against sea.
• - Walking / driving pushing against land.
• - Rockets self contained and pushes against
  its own exhaust gasses.
• Note
• - Rockets gain momentum gradually (rather
  than through a single brief impulse) but
  can be thought
• of as a continuous series of small impulses.
• - As their mass decreases during flight,
  the resultant velocity is more difficult to
  calculate.