As mentioned in the previous part
of this lesson, momentum is a commonly used term in
sports. When
a sports announcer says that a team has the momentum they
mean that the team is really on the move and is
going to be hard to stop. An object with momentum
is going to be hard to stop. To stop such an object, it
is necessary to apply a force
against its motion for a given period of time. The
more momentum which an object has, the harder that it is
to stop. Thus, it would require a greater amount of force
or a longer amount of time (or both) to bring an object
with more momentum to a halt. As the force acts upon the
object for a given amount of time, the object's velocity
is changed; and hence, the object's momentum is
changed.
The
concepts in the above paragraph should not seem like
abstract information to you. You have observed this a
number of times if you have watched the sport of
football. In football, the defensive players apply a
force for a given amount of time to stop the momentum of
the offensive player who has the ball. You have also
experienced this a multitude of times while driving. As
you bring your car to a halt when approaching a stop sign
or stoplight, the brakes serve to apply a force to the
car for a given amount of time to stop the car's
momentum. An object with momentum can be stopped if a
force is applied against it for a given amount of
time.
A force acting for a given amount of
time will change an object's momentum. Put another way,
an unbalanced force
always accelerates an object - either speeding it up or
slowing it down. If the force acts opposite the
object's motion, it slows the object down. If a force
acts in the same direction as the object's motion, then
the force speeds the object up. Either way, a force will
change the velocity of an object. And if the velocity of
the object is changed, then the momentum of the object is
changed.
These concepts are merely an outgrowth
of Newton's second
law as discussed in an earlier unit. Newton's second
law (Fnet=m*a) stated that the acceleration of
an object is directly proportional to the net force
acting upon the object and inversely proportional to the
mass of the object. When combined with the definition of
acceleration (a=change in velocity/time), the following
equalities result.
If both sides of the above equation are multiplied by
the quantity t, a new equation results.
This equation is
one of two primary equations to be used in this unit. To
truly understand the equation, it is important to
understand its meaning in words. In words, it could be
said that the force times the time equals the mass times
the change in velocity. In physics, the quantity
Force*time is known as the impulse. And since the
quantity m*v is the momentum, the quantity m*"Delta "v
must be the change in momentum. The equation really says
that the
Impulse = Change in
momentum
One
focus of this unit is to understand the physics of
collisions. The physics of collisions are governed by the
laws of momentum; and the first law which we discuss in
this unit is expressed in the above equation. The
equation is known as the
impulse-momentum change
equation. The law can be
expressed this way:
In a collision, an
object experiences a force for a specific amount of
time which results in a change in momentum (the
object's mass either speeds up or slows down). The
impulse experienced by the object equals the change in
momentum of the object. In equation form, F * t = m *
Delta v.
In a collision,
objects experience an impulse; the impulse causes (and is
equal to) the change in momentum. Consider a football
halfback running down the football field and encountering
a collision with a defensive back. The collision would
change the halfback's speed (and thus his momentum). If
the motion was represented by a ticker
tape diagram, it might appear
as follows:
At approximately the tenth dot
on the diagram, the collision occurs and lasts for a
certain amount of time; in terms of dots, the collision
lasts for approximately nine dots. In the
halfback-defensive back collision, the halfback
experiences a force which lasts for a certain amount of
time to change his momentum. Since the collision causes
the rightward-moving halfback to slow down, the force on
the halfback must have been directed leftward. If the
halfback experienced a force of 800 N for 0.9 seconds,
then we could say that the impulse was 720 N*s. This
impulse would cause a momentum change of 720 kg*m/s. In a
collision, the impulse experienced by an object is always
equal to the momentum change.
Now consider a collision of a tennis
ball with a wall. Depending on the physical properties of
the wall (its elastic nature), the speed at which the
ball rebounds from the wall upon colliding with it will
vary. The diagrams below depict the changes in velocity
of the same ball. For each representation (vector
diagram, v-t graph, and ticker tape pattern), indicate
which case (A or B) has the greatest change in velocity,
greatest acceleration,
greatest momentum change, and
greatest impulse. Support each
answer.
Observe
that each of the collisions above involved the rebound of
a ball off a wall. Observe that the greater the
rebound effect, the greater the acceleration,
momentum change, and impulse. A rebound is a special type
of collision involving a direction change; the result of
the direction change is large velocity change. On
occasions in a rebound collision, an object will maintain
the same or nearly the same speed as it had before the
collision. Collisions in which objects rebound with the
same speed (and thus, the same momentum and kinetic
energy) as they had prior to the collision are known as
elastic collisions.
In general, elastic collisions are characterized by a
large velocity change, a large momentum change, a large
impulse, and a large force.
Use the impulse-momentum change
principle to fill in the blanks in the following rows of
the table. As you do, keep these three major truths in
mind:
the impulse experienced by an object is the
force*time
the momentum change of an object is the
mass*velocity change
the impulse equals the momentum change
Depress the mouse on the "pop-up" menus to view
answers.
Force
(N)
time
(s)
Impulse
(N*s)
Mom.
Change
(kg*m/s)
Mass
(kg)
Vel.
Change
(m/s)
1.
0.010
10
-4
2.
0.100
-40
10
3.
0.010
-200
50
4.
-20 000
-200
-8
5.
-200
1.0
50
There are a few observations which can be made in the
above table which relate to the computational nature of
the impulse-momentum change theorem. First, observe that
the answers in the table above reveal that the third and
fourth columns are always equal; that is, the impulse is
always equal to the momentum change. Observe also that
the if any two of the first three columns are known, then
the remaining column can be computed; this is true
because the impulse=force*time. Knowing two of these
three quantities allows us to compute the third quantity.
And finally, observe that knowing any two of the last
three columns allows us to compute the remaining column;
this is true since momentum change = mass*velocity
change.
There
are also a few observations which can be made which
relate to the qualitative nature of the impulse-momentum
theorem. An examination of rows 1 and 2 show that force
and time are inversely proportional; for the same mass
and velocity change, a tenfold increase in the time of
impact corresponds to a tenfold decrease in the force of
impact. An examination of rows 1 and 3 show that mass and
force are directly proportional; for the same time and
velocity change, a fivefold increase in the mass
corresponds to a fivefold increase in the force required
to stop that mass. Finally, an examination of rows 3 and
4 illustrate that mass and velocity change are inversely
proportional; for the same force and time, a twofold
decrease in the mass corresponds to a twofold increase in
the velocity change.
Check
Your Understanding
Express your understanding of the impulse-momentum
change theorem by answering the following questions.
Depress the mouse on the "pop-up" menu to view the
answers.
1. A 0.50-kg cart (#1) is pulled with a 1.0-N force
for 1 second; another 0.50 kg cart (#2) is pulled with a
2.0 N-force for 0.50 seconds. Which cart (#1 or #2) has
the greatest acceleration? Explain.
Which cart (#1 or #2) has the greatest impulse?
Explain.
Which cart (#1 or #2) has the greatest change in
momentum? Explain.
2. In a phun physics demo, two identical balloons (A
and B) are propelled across the room on horizontal guide
wires. The motion diagrams (depicting the relative
position of the balloons at time intervals of 0.05
seconds) for these two balloons are shown below.
Which balloon (A or B) has the greatest acceleration?
Explain.
Which balloon (A or B) has the greatest final
velocity? Explain.
Which balloon (A or B) has the greatest momentum
change? Explain.
Which balloon (A or B) experiences the greatest
impulse? Explain.
3. Two cars of equal mass are
traveling down Lake Avenue with equal velocities. They
both come to a stop over different lengths of time. The
ticker tape patterns
for each car are shown on the diagram below.
At what approximate location on the diagram (in terms
of dots) does each car begin to experience the
impulse.
Which car (A or B) experiences the greatest
acceleration? Explain.
Which car (A or B) experiences the greatest change in
momentum? Explain.
Which car (A or B) experiences the greatest impulse?
Explain.
4.
The diagram to the right depicts the before- and
after-collision speeds of a car which undergoes a
head-on-collision with a wall. In Case A, the car bounces
off the wall. In Case B, the car "sticks" to the
wall.
In which case (A or B) is the change in velocity the
greatest? Explain.
In which case (A or B) is the change in momentum the
greatest? Explain.
In which case (A or B) is the impulse the greatest?
Explain.
In which case (A or B) is the force which acts upon
the car the greatest (assume contact times are the same
in both cases)? Explain.
5. Rhonda, who has a mass of 60.0 kg,
is riding at 25.0 m/s in her sports car when she must
suddenly slam on the brakes to avoid hitting a dog
crossing the road. She strikes the air bag, which brings
her body to a stop in 0.400 s. What average force does
the seat belt exert on her?
If Rhonda had not been wearing her seat belt
and not had an air bag, then the windshield would have
stopped her head in 0.001 s. What average force would the
windshield have exerted on her?
6. A hockey player applies an average force of 80.0 N
to a 0.25 kg hockey puck for a time of 0.10 seconds.
Determine the impulse experienced by the hockey puck.
7. If a 5-kg object experiences a
10-N force for a duration of 0.1-second, then what is the
momentum change of the object?
This equation is
one of two primary equations to be used in this unit. To
truly understand the equation, it is important to
understand its meaning in words. In words, it could be
said that the force times the time equals the mass times
the change in velocity. In physics, the quantity
Force*time is known as the impulse. And since the
quantity m*v is the momentum, the quantity m*"Delta "v
must be the change in momentum. The equation really says
that the
Impulse = Change in
momentum
When
a sports announcer says that a team has the momentum they
mean that the team is really on the move and is
going to be hard to stop. An object with momentum
is going to be hard to stop. To stop such an object, it
is necessary to apply a 
