Download Momentum and impulse

Momentum and impulse
The momentum-impulse equation
For a body of mass m, moving with velocity v,
momentum  mv
Consider a constant force F acting for a time t on a body of mass m
in the direction of its motion, causing the velocity to increase from u to v.
Then F  ma and v  u  at quickly yields Ft  m(v  u)  mv  mu.
The product of the force and the time for which it acts is called
the impulse of the force and is denoted by the vector J .
impulse  change in momentum
J  Ft  mv  mu
Note : The impulse unit
usually given is N s.
Read Examples 13a, pp.263-266 & Example 13b, p.268
Do Exercise 13a, pp.266-268 & Exercise 13b, pp.268-269
Collision
 Two objects in contact exert equal and opposite forces on each other.
 Since objects are in contact with each other for the same time, they
exert equal and opposite impulses on each other.
 As long as no external force acts on either object, the total momentum of
the two objects remain constant.
The Principle of Conservation of Linear Momentum :
If in a specified direction, no external force affects
the motion of a system, the total momentum in that
direction remains constant.
Note :
 Objects that collide and join together at impact are said to coalesce.
 It is advisable to draw separate "before" and "after" diagrams
when solving problems involving collisions.
Read Examples 13c, pp.270-272
Do Exercise 13c, pp.272-274
Loss of kinetic energy; finding the
impulse
When there is a sudden change in the motion of a system, there is
usually a change in the total kinetic energy of the system.
This is because some mechanical energy is converted into sound energy
or heat energy (or both).
When objects collide, equal and opposite impulses act on the two objects.
When the system is treated as a whole, these impulses cancel and need
not appear in calculations.
The impulse which acts on one object causes the change in the
momentum of that object only.
So, if the magnitude of the impulses is required, it is sufficient to
consider only one of the colliding objects.
Read Examples, pp.274-276
Do Exercise 13d, pp.276-277
Impacts
Many collisions so far have resulted in the objects coalescing at impact.
Such impacts are called inelastic.
However, if a bounce occurs at collision, we have an elastic impact and
the colliding object(s) are said to be elastic.
The simplest examples of inelastic impacts are direct impacts, i.e. an
impact in which the direction of motion just before impact is parallel
to the impulses that act at the instant of collision.
Newton’s Law of Restitution
Experimental evidence suggests that, for two colliding particles, the
separation speed is always the same fraction of the approach speed.
Newton's Law of Restitution :
separation speed  e  approach speed
e is called the coefficient of restitution and it is constant for any two particular
objects; its value depends upon the materials of which the two objects are made.
 0  e 1
 If particles coalesce the separation speed is zero i.e. e  0.
 If the relative speeds are equal, e  1, the particles are said to be
perfectly elastic (there will be no loss in kinetic energy).
Read Examples 14a, pp.280-281; 14b, pp.284-286; 14c, pp.290-292
Collision with a fixed object
Consider a particle of mass m, moving on a smooth horizontal surface with
speed u, towards a fixed block whose face is perpendicular to the direction
of motion of the particle.
When the particle hits the block an impulse J is
exerted on the particle by the block and, if the
impact is elastic, the particle bounces off the block
in the oppsite direction with speed v, say.
If the direction of J is taken as positive,
v  eu
J  mv  (mu )
Note : The conservation of linear momentum is not valid here since the impulse
applied to the particle by the fixed surface is an external impulse.
Do Exercises 14a, pp.281-283; 14b, pp.287-289; 14c, pp.293-295
Oblique impact with a fixed object
Consider a sphere travelling on a horizontal surface colliding with a
vertical wall, the direction of its velocity making an angle  with the wall.
The impulse exerted on the sphere is perpendicular to
the wall and causes a change in the momentum of the
sphere in that direction only; it does not affect the
momentum parallel to the wall.
The component of the velocity
perpendicular to the wall may be
changed whereas the component
parallel to the wall must remain
unchanged.
Note : Newton's restitution law here applies to the component of velocity
perpendicular to the surface before and after impact.
Read Example 14d, p.297
Do Q1, Q2, Q7, pp.300-301