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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 Δt (units: N.s) - a vector
where F is the force and Δt is the time of action.
• By Newton’s 2nd law (F = m.a) we determined:

Impulse = F. Δt = m. Δv
or Impulse = Change in momentum (ΔP)
impulse
(F. Δt)
m
momentum change
(ΔP = m. Δv)
Result: Large impulses cause large changes in motion!
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 it’s the change in momentum of an object that is
important as this equals the impulse:
ΔP = m. Δv = impulse
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
Δv
m
• Impulse = m. Δv
F. Δt
• A new way of looking at Newton’s 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. Δt = 4000 x 0.5 = 2000 N.s
Thus: Change in momentum: = 2000 Ns = m. Δv
or Δv = 2000
50 = 40 m/s ( ≈ 140 km/hr)
Note: When the initial velocity is zero:
• change in momentum = objects momentum
• change in velocity = objects velocity
Ex: Ball’s 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.
ΔP
P2=m.v
P1=m.v
impulse
before
during
after
3. The total change in momentum ΔP is:
ΔP = P2 – P1 = m.v – (- m.v)
(As P1 opposite to P2)
ΔP = 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.)
Summary:
• Rewriting Newton’s 2nd law shows that:
Impulse = Change in momentum (ΔP)
Impulse = F. Δt = m. Δv
• Large impulses produce large changes in
momentum -resulting in large velocity changes.
Where:
Impulse = F x Δt (units: N.s) - a vector
Momentum: P = m.v (units: kg.m/s) - a vector.
Conservation of Momentum
• A new principle for studying collisions which results from
Newton’s 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
other…they 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 Δt.
• By Newton’s 3rd law, an equal and opposite force ‘F’ acts
back (remember forces occur in pairs).
Conservation of Momentum
• As Δt is the same for both forces,
the impulses they produce
(= F.Δt) are the same magnitude
but in opposite directions.
• By Newton’s 2nd law:
v1
- impulse
v2
impulse
Impulse = Change in momentum (ΔP = m. Δv)
- Thus the change in momentum experienced by both
objects must be the same…but 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).
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…
Example: Custard pie fights!
mc = 1 kg
mt = 100 kg
vc = -10 m/s
vt = 0.5 m/s
positive direction
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)!
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”
- rocket propulsion…
- firing a gun
• We have already looked at what happens to the ice skaters
motion (using Newton’s 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!
Firing a Gun
(initial momentum = zero)
Momentum of bullet = Momentum of gun
m1v1 = - m2v2
• Mass of bullet is small but its velocity is high…creating 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 one…for 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.
Collisions
• Two main types: Elastic and Inelastic…
• Different kinds of collisions produce different
results…e.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?
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)
v = 10 m/s
2x104 kg
3 stationary trucks
104 kg
5x103 kg
15x103 kg
Before: System momentum: mv = 2x104x10 = 2x105 kg.m/s
After: Final system mass = (20+10+5+15)x103 kg = 50x103 kg
As final momentum = initial momentum
Pfinal
2
x105
= 4 m/s
vfinal =
=
4
total mass
5x10
• Thus total momentum of system has remained constant but the
colliding truck’s velocity has reduced (i.e. momentum shared).
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!
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.
• 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:
Question: What happens to
P1
red ball and cue ball?
cue ball
(no spin)
P2
• 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.
• It’s a fact…try it for yourself!!!
Impulse
 Impulse is the average force acting on an object
multiplied by its time interval of action.
Impulse = F . Δt (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 (Δt) the
larger the impulse. ( Impulse is therefore a measure of the
overall effect of the force.)
However: Impulse = F. Δt = m. Δv
• So an impulse causes a change in velocity (Δv) in
magnitude and direction.
• In Newton’s words the product “m.Δv” is the change in the
“quantity of motion”.
• We now term this the change in momentum.
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.