LEARNING GOALS
• The meaning of the momentum of a particle, and how the impulse of the net force
acting on a particle causes its momentum to change.
• The conditions under which the total momentum of a system of particles is constant
(conserved).
• How to solve problems in which two bodies collide with each other.
• The important distinction among elastic, inelastic, and completely inelastic collisions.
?
There are many questions
involving forces that cannot
be answered by directly
applying Newton’s second
law.
For example, in a car crash, what determines which way the wreckage moves
after the collision?
In playing pool, how do you decide how to aim the cue ball in order
to knock the eight ball into the pocket?
When a meteorite collides with the earth, how much of the
meteorite’s kinetic energy is released in the impact?
A common theme of all these questions is that they involve forces
about which we know very little:
• the forces between the car and the moving van,
• between the two pool balls, or
• between the meteorite and the earth.
We don’t have to know anything about these forces
to answer questions of this kind!
Our approach uses two
momentum and impulse
new concepts; ,
and a new
conservation of momentum.
conservation law,
Within the domain of Newtonian mechanics, conservation
of momentum enables us to analyze many situations that
would be very difficult if we tried to use Newton’s laws
directly.
Among these are collision problems, in which two bodies
collide and can exert very large forces on each other for a
short time.
Momentum and Impulse
Newton’s Second Law in Terms of Momentum:
Consider a particle of constant mass m.
Newton’s second law for this particle:
the product of the
particle’s mass and
velocity.
the momentum, or
We’ll call this
linear momentum,
combination of the particle.
Keep in mind that momentum is a vector quantity with the
same direction as the particle’s velocity :
We often express the momentum of a particle
in terms of its components.
*which we also call the x-momentum, y-momentum, and z-momentum!
The net force (vector sum of all forces) acting on a
particle equals the time rate of change of momentum of
the particle.
*This is the form in which Newton originally stated his second law
(although he called momentum the “quantity of motion”).
A rapid change in momentum requires a large net force,
while a gradual change in momentum requires less net force.
This principle is used in the design of automobile safety devices
such as air bags.
The Impulse–Momentum Theorem
Let’s go back to Newton’s second law
as restated in terms of momentum:
The change in momentum of a particle during a
time interval equals the impulse of the net force
that acts on the particle during that interval.
The impulse–momentum theorem also holds when forces
are not constant:
Momentum and Kinetic Energy Compared
We’re given enough information to determine the initial and final
values of the ball’s momentum, so we can use the impulse–
momentum theorem to find the impulse.
We’ll then use the definition of impulse to determine the average
force.
The force that the wall exerts on the ball must have such a large
magnitude (2000 N, equal to the weight of a 200-kg object)
to change the ball’s momentum in such a short time.
Other forces that act on the ball during the collision are
comparatively weak; for instance, the gravitational force is only
3.9 N.
Thus, during the short time that the collision lasts, we can ignore
all other forces on the ball.
Note that the 2000-N value we
calculated is the average
horizontal force that the wall
exerts on the ball during the
impact. It corresponds to the
horizontal line in Fig.
The horizontal force is zero before
impact, rises to a maximum, and
then decreases to zero when the
ball loses contact with the wall.
If the ball is relatively
rigid, like a baseball or
golf ball, the collision lasts
a short time and the
maximum force is large,
as in the blue curve in Fig.
If the ball is softer, like a
tennis ball, the collision
time is longer and the
maximum force is less, as
in the orange curve in Fig.
The ball moves in two dimensions, so we must treat momentum and
impulse as vector quantities.
We take the x-axis to be horizontally to the right and the y-axis to be vertically
upward.
Use impulse–momentum theorem!
We need velocity components!
Conservation of Momentum
Each particle exerts a force on the other;
according to Newton’s third law, the two
forces are always equal in magnitude and
opposite in direction.
Hence, the impulses that act on the two
particles are equal and opposite, and
the changes in momentum of the two
particles are equal and opposite.
Let’s consider first an idealized
system of two bodies that
interact with each other but not
with anything else.
For any system, the forces that the particles of the system exert on
each other are called internal forces.
Forces exerted on any part of the system by some object outside it
are called external forces.
There are no
external forces; when this is
the case, we have an
isolated system.
Principle of conservation of momentum:
If the vector sum of the external forces on a system is zero, the
total momentum of the system is constant.
The bullet and rifle have equal and opposite final
momenta thanks to Newton’s third law: They experience equal
and opposite interaction forces that act for the same time, so the
impulses are equal and opposite.
But the bullet travels a much greater distance than the rifle during
the interaction. Hence the force on the bullet does more work than
the force on the rifle, giving the bullet much greater kinetic energy
than the rifle.
*Forces between the
bodies are
conservative
*Bodies stick
together and move
as one body
How to measure the speed of a bullet?
The Ballistic
Precision
Chronograph
provides accurate velocity
measurements across a
wide range of shooting
conditions.
