PHYS 2108 Lecture Notes - Lecture 6: Momentum, Headon, Lab Partners
6-1
LAB 6:
Conservation of Linear Momentum
Ituitiel, hih of the folloig has oe otio, a osuito o a sei-truck travelling at the same
velocity? Which is more difficult to slow down? Which would win in a collision and which is more
destructive? I askig hih ojet otais oe otio, e see to ko the ase ee though
both are travelling with the same velocity, and this means the velocity alone appears not to be enough
to ase the uestio. Whatee this ualit of otio is, it appears also to depend upon the
ojet’s mass.
You may be tempted to say that the kinetic energy of the objects can see as a gauge of the ojet’s
otaied otio, ut consider an exploding ball. Before the explosion, there is no motion. After the
explosion, fragments may fly off in many directions. A sste’s ete of ass a ol e oed he
acted upon by an outside force, and because the eplosio is osideed a iteal foe, the
fagets’ combined center of mass is unmoved. Using kinetic energy (a scalar quantity) is
iappopiate fo desiig the otio a ojet possesses eause thee’s o a to add up
contributions from the fragments in a way that gives us zero for the combined center of mass. We need
something else that is a vector.
The quality of otio we have described is called momentum, specifically linear momentum. It is
given the symbol
, it is a vector quantity, and it is a conserved quantity in physics. If we allow to
epeset a ojet’s ass ad
to epeset its eloit, the the ojet’s oetu is
Eqn. 6-1:
Linear Momentum
The momentum vector’s direction is the same as the velocity vector’s. The oetu’s agitude is
the product of (mass · velocity), which gives us the unit of momentum, kg · m/s. Unlike Newtons (kg ·
m/s2) for force or Joules (N · m or kg · m2/s2) for energy, the unit for momentum has no special name.
In the case of the mosquito and semi-truck mentioned above, if we were to slow them both to a halt in
the same amount of time, the semi-truck would require much greater force due to its greater
momentum. Similarly, it would have a more dominant effect than the mosquito in a head-on collision.
The faster-moving or more-massive object is not always the dominant one! A speeding low-mass car
that did’t see the ed light ill do uh oe daage tha a high-mass semi-truck slowly inching
forward at an intersection. A light high-velocity spitwad will do much less damage than the heavy
slowly-lumbering bully who shot it. One must always consider the product of mass and velocity over
either one of them alone.
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6-2
The momentum of an object a ol e haged a eteal foe. This is i fat Neto’s faous
Second Law of Motion; more specifically, the net rate of change of momentum is proportional to the
net force applied.
This defiitio of Neto’s “eod Law is more general than the version you may have first learned
because it allows for situations where the mass might not be constant while the velocity changes (e.g.,
rocket fuel consumed in flight, exotic circumstances involving Special Relativity). In cases where the
mass is constant, this equation takes a more familiar form.
Collisions
Momentum is a conserved quantity in the absence of an external force. That means in a collision
between two (or more objects), the total momentum before the interaction is equal to the total
momentum after the interaction—the Law of Conservation of Momentum.
Eqn. 6-2:
Momentum Conservation
The motion of two objects involved in a collision can be predicted and examined by considering the
conservation of momentum. There are two varieties of collisions.
Elastic collisions: Momentum and Kinetic Energy are conserved during the collision.
Eqn. 6-3:
Conservation of KE
In perfectly elastic collisions, the objects perfectly transfer kinetic energy with none of it
converted into other forms such as heat, sound and physical deformation. A nearly perfectly
elastic collision is similar to two billiard balls wherein one stops completely after colliding with a
stationary other while the other leaves the collision with the same speed as the first.
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6-3
Inelastic collisions: Momentum only is conserved. Some Kinetic Energy is lost to other forms.
Because the kinetic energy is not conserved before and after the collision, using it to analyze the
collision is not usually helpful. Remember that this does not violate the law of conservation of
energy to say that KE is not conserved. The total amount of energy is the same; it’s just ee
changed from KE to thermal heat, deformation of the objects, sound, or various other forms.
In a perfectly inelastic collision, there is a certain theoretial aiu of osuale kieti
energy that must be lot. This osuale KE is taken mostly by the sticking together of
objects involved, a sure sign of a perfectly inelastic collision.
During any collision the objects are deformed slightly in order to generate a force on one
aothe. I a elasti ollisio the to shapes spig ak to thei oigial shape. I a
inelastic collision they do not. Two billiard balls collide almost perfectly elastically; two balls of
clay do not.
Most real-life collisions are not perfectly elastic or inelastic. “oe eeg is alas lost fo a elasti
collision, and some of the consumable kinetic energy always escapes consumption in an inelastic
collision.
Explosions
A special momentum-oseig eese ollisio situatio ioles a ojet breaking up into
fragments. For an object at rest exploding into two pieces, each piece must have equal and opposite
momenta in order to sum to zero. The same applies to an object already in motion. Any extra
oetu gaied the foad piee ust e deduted fo the rearward piee suh that the
net momentum for the center of mass remains unchanged.
Ee though the fagets’ oeta add to zeo for an exploding object initially at rest, the fact that
they have velocities mean they have kinetic energies. Those energies must have originated from
somewhere. A stored eplosio potetial eeg is converted to kinetic energy and released to the
fragments. This potential energy can come from many sources. It can be chemical, nuclear, electrical or
as simple as the stored energy in a compressed spring.
A real-world explosion will not perfectly convert the stored explosion energy into KE with 100%
efficiency. Some will be lost to other forms. Consider an exploding festival ball at a fireworks show. The
eplosio does’t ol satte the fagets—it also releases heat, a concussive boom and glowing
light.
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