SPH4U1: ENERGY & MOMENTUM LESSON 7 – Conservation of Momentum
Law of Conservation of Linear Momentum
If the net force acting on a system of interacting objects is zero, then the linear
momentum of the system before the interaction equals the linear momentum of the
system after the interaction.
NOTE: A system represents all of the objects involved in a collision. If the net external
force on all objects as a group is zero, we say the system is isolated or closed.
Let’s use Newton’s third law to come up with an equation for this:
Therefore, for 2 solid objects colliding, the conservation of momentum can be written as




m1v1i  m2 v2i  m1v1 f  m2 v2 f
This can easily be split into components so that just the horizontal and/or vertical
aspects can be analyzed.
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


m1v1ix  m2 v 2ix  m1v1 fx  m2 v 2 fx
and
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


m1v1iy  m2 v 2iy  m1v1 fy  m2 v 2 fy
EX 1: A shell of mass 7.0kg leaves the muzzle of a cannon with a horizontal velocity of
490m/s [right]. Find the recoil velocity of the cannon if it’s mass if 700kg.
SPH4U1: ENERGY & MOMENTUM LESSON 7 – Conservation of Momentum
ELASTIC and INELASTIC COLLISIONS
Conservation of momentum can be applied to many collisions but there are two basic
types of two-particle collisions that always conserve momentum. The typical situation in
momentum conservation involves two particles in the initial system with one or both
having a velocity. These two particles collide where again only internal forces act and
the particles separate with certain final velocities. Conservation of momentum enables
us to relate the final velocities to the initial velocities. The two basic types of collisions
are:
1) Elastic Collisions
2) Inelastic Collisions
1) Elastic Collisions
An elastic collision is one in which the total kinetic energy of the two particles is the
same after the collision as it was before the collision. Examples of elastic collisions are
those between billiard balls, between masses and springs, and those involving rubber or
tennis balls. For elastic collisions one can write not only the momentum conservation
equation, but also a kinetic energy conservation equation:




m1v1i  m2 v2i  m1v1 f  m2 v2 f (Momentum conserved)
1  2 1  2 1  2 1  2
m1v1i  m2 v2i  m1v1 f  m2 v2 f (Kinetic Energy conserved)
2
2
2
2
By combining these two equations one can achieve the general result that
v1i  v 2i  v1 f  v 2 f

The relative velocity of approach is the negative of the relative velocity of separation.
SPH4U1: ENERGY & MOMENTUM LESSON 7 – Conservation of Momentum
2) Inelastic and Perfectly Inelastic Collisions
If there are very strong frictional and deformation forces, then kinetic energy will no
longer be conserved and instead we will have an inelastic collision. The limiting case of
an inelastic collision is one in which the two particles fuse during the collision, and travel
together afterwards with the same final velocity.
v1 f  v 2 f  v f
(Perfectly Inelastic Collision)



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m1v1i  m2v2i  m1v f  m2v f
m1 v1i  m2 v 2i
vf 
m1  m2
(Particles with same final velocity)
(Perfectly Inelastic Collision)
EX 2: Two billiard balls have velocities of 2.0m/s and -0.5 m/s before they meet in a head
on collision. What are their final velocities?
NOTE: In equal mass elastic collisions in one dimension, the masses simply exchange
velocities. In equal mass, one dimensional elastic collisions with the first particle at rest,
the second particle stops and the first particle goes forward with the original velocity of
the second particle.
SPH4U1: ENERGY & MOMENTUM LESSON 7 – Conservation of Momentum
EX 3:
A Cadillac with a mass of 1800 kg, while stopped at a traffic light, is rear ended by a
Volkswagen with a mass of 900 kg traveling at 20 m/s. After the collision both cars are
completely entangled, and slide into the intersection. What is their velocity after the
collision?
HMWK: Pg. 243 #4-8
Pg. 245 #6-10,
Pg. 253 #4-7