Unit 5 of Physics in the Leveling course at Yachay Tech University (Urcuquí, Ecuador). Semester October 2014-March 2015

1. UNIT 5. Impulse and momentum
Nivelación. Física

2. 5.1 Momentum
• Linear momentum is a vectorial magnitude:
– p = mv
• It is considered the difficulty of managing a
particle at rest.
• Momentum is a key concept in Physics.
• Kinetic energy helps the system analysis with
momenta:
– E = ½ mv2 = p2/2m.

3. 5.2 Impulse
• Impulse is a force applied during a time
interval:
– I = F Dt
• Moreover, net impulse quantifies the change
of momentum of the system:
– I = Dp
• Average force can be calculated with impulse,
time interval and distance.

4. 5.3 Center of mass
• An object moving along the
space can be described by the
movement of the center of mass
(representative of the whole
system) and the relative
movement of particles around it.
• We can describe the movement
of a system by its center of mass:
a particle with the total mass of
the system.
– rcm Si mi = Si miri

5. 5.3 Center of mass
• Hint! Center of mass is close to the heaviest
mass of the system.
• It is possible to apply easy calculations of
kinematics, dynamics and potential energy
using the center of mass of a complex system.
• According to this, velocity and acceleration of
a center of mass can be described by:
– vcm Si mi = Si mivi
– acm Si mi = Si miai

6. 5.3 Center of mass
• In the frame of center of mass all particle positions are
referred to the center of mass location, so the
transformation between the laboratory frame and the
CoM frame is:
– ri = rcm + r’i

7. 5.3 Center of mass
• Indeed, the net external force applied to a system
only changes the acm, not the relative
accelerations of each particle.
• Internal forces between particles are
compensated by the Newton’s 3rd law, not
affecting to the global system.
– Fext
net = Si Fext
i = acm Si mi
• Furthermore, the momentum of a system is
described by vcm:
– psyst = vcm Si mi

8. 5.3 Center of mass
• In a similar fashion, the kinetic energy of a
system can be described by the kinetic energy
of the center of mass and the kinetic energy of
the particles relative to the CoM (i.e., center
of mass).
• Ec = Ec
cm + Ec
rel = ½ Si mi vcm
2 + ½ Si mivi
2

9. 5.4 Newton’s laws in terms of
momentum
• First law: if external forces are zero, the
momentum of a system is constant.
• Second law: a force acting on a system changes in
one instant its momentum.
• Third law: when a system exerts a temporal
variation of momentum on other system, this one
exerts the same temporal variation of
momentum, with the same modulus and
direction, but in reverse.

10. 5.5 Conservation of momentum
• According to the Newton’s 1st law,
momentum does not change if there are no
external forces.
– Si Fext
i = 0 and acm = 0.
• In other words, an isolated system conserves
its total momentum:
– psyst = cst. and vcm = cst.

11. 5.5 Conservation of momentum
• It is a very important conservation law in
Physics! In general, internal forces are not
conservatives, so the conservation of
mechanical energy is not longer valid.
– pstart = pfinish, that is, Si mivs
i = Si mivf
i
• Momentum could be only conserved in one
space direction, so a previous analysis must be
carefully done.

12. 5.6 Elastic and inelastic collisions
• Collision is a strong interaction in a short lapse
of time between particles.
• External forces are not considered during
collisions.
• During collision, internal forces are relevant
and the total momentum of the system is
conserved.

13. 5.6 Elastic and inelastic collisions
• There are three main types of collision:
– Elastic: the total kinetic energy of the system is
the same before and after collision.
– Inelastic: after the collision, the total kinetic
energy changes in the system.
– Perfectly inelastic: the kinetic energy is
completely lost, transformed in heat or internal
energy. Particles are melted in a new massive
particle.

14. 5.6 Elastic and inelastic collisions
• A head-on collision is a 1-D
problem. Speeds after collision
could be positive or negative,
depending on the conservation
of momentum.
• The conservation law of the
momentum in a collision
developed in a plane must be
verified in each axis.
• Collisions in the space are a bit
complicated, because there are
many variables to consider, like
the interaction force involved
and the vertical distance
between the particle’s centers
(impact parameter).
• However, the whole process is
confined in a plane.

15. 5.6 Elastic and inelastic collisions
• It is worth to mention that when two identical
masses (but one at rest) collide in an elastic
fashion and b ≠ 0, final velocities configure a
right angle.

16. 5.6 Elastic and inelastic collisions
• In an elastic collision, kinetic energy is
conserved.
• Moreover, in a head-on collision between two
particles, speeds verify that
– v2
f – v1
f = -(v2
s – v1
s)

17. 5.6 Elastic and inelastic collisions
• In a perfectly inelastic
collision final speeds are
the same, i.e. vcm.
• Remember that in this
situation mi are combined
in one mass: Si mi.
• Kinetic energy before and
after collision must be
considered also in order to
simplify calculation.