Download momentum

Momentum
A heavy truck is harder to stop than a
small car moving at the same speed.
We state this fact by saying that the truck
has more momentum than the car.
By momentum we mean inertia in motion.
More specifically momentum is defined as
the product of the mass of an object and
its velocity.
Which we abbreviate mv.
Moving Object
A moving object can have a large momentum if either its
mass or its velocity is large, or if both its mass and its
velocity are large.
The truck has more momentum than the car moving at
the same speed because it has a greater mass.
We can see that a huge ship moving at a small speed
can have a large momentum, as can a small bullet
moving at a high speed.
And, of course, a huge object moving at a high speed,
such as a massive truck rolling down a steep hill with no
brakes, has a huge momentum,
whereas the same truck at rest has no momentum at
all—because the v part of mv is zero.
Why are the engines of a supertanker
normally cut off 25 km from port?
Force and Momentum
Changes in momentum may occur when there is
either a change in the mass of an object, a
change in velocity, or both.
If momentum changes while the mass remains
unchanged, as is most often the case, then the
velocity changes.
Acceleration occurs.
And what produces an acceleration?
The answer is a force.
The greater the force that acts on an object, the
greater will be the change in velocity and,
hence, the change in momentum.
Time and Momentum
But something else is important in changing
momentum: time—how long a time the force
acts.
Apply a force briefly to a stalled automobile, and
you produce a small change in its momentum.
Apply the same force over an extended period of
time, and a greater change in momentum
results.
A long sustained force produces more change in
momentum than the same force applied briefly.
Force and Time-Impulse
So for changing the momentum of an object,
both force and the time during which the force
acts are important. We name the product of
force and this time interval impulse.
Whenever you exert a net force on something,
you also exert an impulse.
The resulting acceleration depends on the net
force;
the resulting change in momentum depends on
both the net force and the time during which that
force acts.
Check Yourself
1. Which has more momentum, a 1-ton car
moving at 100 km/h or a 2-ton truck
moving at 50 km/h?
2. Does a moving object have impulse?
3. Does a moving object have
momentum?
Impulse Changes Momentum
Impulse changes momentum in much the same
way that force changes velocity.
The relationship of impulse to change of
momentum comes from Newton's second law (a
= F/m).
The time interval of impulse is “buried” in the
term for acceleration (change in velocity/time
interval).
Rearrangement of Newton's second law gives
“force multiplied by the time during which it acts
equals change in momentum.”
Impulse-Momentum
The impulse-momentum relationship helps us to
analyze many examples in which forces act and
motion changes.
Sometimes the impulse can be considered to be
the cause of a change of momentum.
Sometimes a change of momentum can be
considered to be the cause of an impulse. It
doesn't matter which way you think about it.
The important thing is that impulse and change
of momentum are always linked.
Here we will consider some ordinary examples
in which impulse is related to -
Case 1-increasing momentum
If you wish to increase the momentum of something as much as
possible, you not only apply the greatest force you can, you also
extend the time of application as much as possible.
Long-range cannons have long barrels. The longer the barrel, the
greater the velocity of the emerging cannonball or shell.
Why? The force of exploding gunpowder in a long barrel acts on the
cannonball for a longer time. This increased impulse produces a
greater momentum.
Of course the force that acts on the cannonball is not steady—it is
strong at first and weaker as the gases expand.
In this and many other cases the forces involved in impulses vary
over time. The force that acts on the golf ball, for example, increases
rapidly as the ball is distorted and then diminishes as the ball comes
up to speed and returns to its original shape.
When we speak of impact forces in this chapter, we mean the
average force of impact.
Case 2-Decreasing Momentum
Over a Long Time
Imagine you are in a car out of control, and you have a choice of
slamming into either a concrete wall or a haystack.
You needn't know much physics to make the better decision, but
knowing some physics helps you to understand why hitting
something soft is entirely different from hitting something hard.
In the case of hitting either the wall or the haystack, your momentum
will be decreased by the same amount, and this means that the
impulse needed to stop you is the same.
The same impulse means the same product of force and time—not
the same force or the same time.
You have a choice. By hitting the haystack instead of the wall, you
extend the time of impact—you extend the time during which your
momentum is brought to zero.
The longer time is compensated by a lesser force. If you extend the
time of impact 100 times, you reduce the force of impact by 100.
So whenever you wish the force of impact to be small, extend the
time of impact.
A large change in momentum in a long time
requires a small force.
A wrestler thrown to the floor tries to extend his
time of arrival on the floor by relaxing his
muscles and spreading the crash into a series of
impacts as foot, knee, hip, ribs, and shoulder
fold onto the floor in turn. The increased time of
impact reduces the force of impact.
A large change in momentum in a short
time requires a large force.
The boxer's jaw provides an impulse that reduces the
momentum of the punch.
(a) The boxer is moving away when the glove hits,
thereby extending the time of contact. This means the
force is less than if the boxer had not moved.
(b) The boxer is moving into the glove, thereby lessening
the time of contact. This means that the force is greater
than if the boxer had not moved.
Case 3- Decreasing Momentum
Over a Short Time
When boxing, move into a punch instead of away, and
you're in trouble.
Likewise if you catch a high-speed baseball while your
hand moves toward the ball instead of away upon
contact.
Or when out of control in a car, drive it into a concrete
wall instead of a haystack and you're really in trouble. In
these cases of short impact times, the impact forces are
large.
Remember that for an object brought to rest, the impulse
is the same, no matter how it is stopped.
But if the time is short, the force will be large.
The idea of short time of
contact explains how a karate
expert can sever a stack of bricks
with the blow of her bare hand .
She brings her arm and
hand swiftly against the bricks
with considerable momentum.
This momentum is quickly reduced
when she delivers an impulse to the bricks.
The impulse is the force of her hand
against the bricks multiplied
by the time her hand makes contact
with the bricks.
By swift execution she makes the
time of contact very brief and
correspondingly makes
the force of impact huge.
If her hand is made to bounce upon impact,
the force is even greater.
Check Yourself
1. If the boxer is able to make the duration of impact
three times as long by riding with the punch, by how
much will the force of impact be reduced?
2. If the boxer instead moves into the punch such as to
decrease the duration of impact by half, by how much
will the force of impact be increased?
3. A boxer being hit with a punch contrives to extend time
for best results, whereas a karate expert delivers a force
in a short time for best results. Isn't there a contradiction
here?
4. When does impulse equal momentum?
Bouncing
You know that if a flower pot falls from a shelf onto your
head, you may be in trouble.
And whether you know it or not, if it bounces from your
head, you may be in more serious trouble.
Impulses are greater when bouncing takes place.
This is because the impulse required to bring something
to a stop and then, in effect, “throw it back again” is
greater than the impulse required merely to bring
something to a stop.
Suppose, for example, that you catch the falling pot with
your hands. Then you provide an impulse to catch it and
reduce its momentum to zero.
If you were to then throw the pot upward, you would
have to provide additional impulse.
So it would take more impulse to catch it and throw it
back up than merely to catch it.
The same greater impulse is supplied by your head if the
pot bounces from it.
The Pelton Wheel
An interesting application of the greater impulse that
occurs when bouncing takes place was employed with
great success in California during the gold rush days.
The water wheels used in gold-mining operations were
ineffective.
A man named Lester A. Pelton saw that the problem had
to do with their flat paddles.
He designed curved-shape paddles that would cause the
incident water to make a U-turn upon impact—to
“bounce.”
In this way the impulse exerted on the water wheels was
greatly increased. Pelton patented his idea and made
more money from his invention, the Pelton wheel, than
most gold miners made from gold.
Check Yourself
Check Yourself
1. How does the force that the woman
exerts on the bricks compare with the
force exerted on her hand?
2. How will the impulse resulting from the
impact differ if her hand bounces back
upon striking the bricks?
External Impulse Changes
Momentum
Recall what Newton's second law tells us: If we
want to accelerate an object, we must apply a
force.
We say much the same thing in this chapter
when we say that to change the momentum of
an object we must apply an impulse.
In any case, the force or impulse must be
exerted on the object or any system of objects
by something external to the object or system.
Internal forces don't count.
Internal Forces do not Change
Momentum
For example, the molecular forces within a
baseball have no effect on the momentum of the
baseball, just as a person sitting inside an
automobile pushing against the dashboard, with
the dashboard pushing back, has no effect in
changing the momentum of the automobile.
This is because these forces are internal forces,
that is, forces that act and react within the
systems themselves.
An outside, or external, force acting on the
baseball or automobile is required for a change
in momentum.
If no external force is present, then no change in
momentum is possible.
Conservation of Momentum
When a bullet is fired from a rifle, the forces
present are internal forces.
The total momentum of the system comprising
the bullet and rifle therefore undergoes no net
change.
By Newton's third law of action and reaction, the
force exerted on the bullet is equal and opposite
to the force exerted on the rifle.
The forces acting on the bullet and rifle act for
the same time, resulting in equal but oppositely
directed impulses and therefore equal and
oppositely directed momenta (the plural form of
momentum).
Conservation of Momentum
The recoiling rifle has just as much momentum
as the speeding bullet.
Although both the bullet and rifle by themselves
have gained considerable momentum, the bullet
and rifle together as a system experience no net
change in momentum.
Before firing, the momentum was zero; after
firing, the net momentum is still zero. No
momentum is gained and no momentum is lost.
The momentum before firing is zero. After
firing, the net momentum is still zero,
because the momentum of the rifle is
equal and opposite to the momentum of
the bullet.
Rifle-Bullet Example
Two important ideas are to be learned from the
rifle-and-bullet example.
The first is that momentum, like velocity, is a
quantity that is described by both magnitude and
direction; we measure both “how much” and
“which direction.”
Momentum is a vector quantity. Therefore,
when momenta act in the same direction, they
are simply added; when they act in opposite
directions, they are subtracted.
Rifle-Bullet Example
The second important idea to be taken from the
rifle-and-bullet example is the idea of
conservation.
The momentum before and after firing is the
same.
For the system of rifle and bullet, no momentum
was gained; none was lost.
When a physical quantity remains unchanged
during a process, that quantity is said to be
conserved.
We say momentum is conserved.
Check Yourself
1. A high-speed bus and an innocent bug have a
head-on collision. The sudden change of
momentum for the bug spatters it all over the
windshield. Is the change in momentum of the
bus greater, less, or the same as the change in
momentum experienced by the unfortunate bug?
2. The Starship Enterprise is being pursued in
space by a Klingon warship moving at the same
speed. What happens to the speed of the
Klingon ship when it fires a projectile at the
Enterprise? What happens to the speed of the
Enterprise when it returns the fire?
Collisions
Momentum is conserved in collisions—that is, the net
momentum of a system of colliding objects is unchanged
before, during, and after the collision.
This is because the forces that act during the collision
are internal forces—forces acting and reacting within the
system itself.
There is only a redistribution or sharing of whatever
momentum exists before the collision.
In any collision, we can say
Net momentum before collision = net momentum
after collision.
This is true no matter how the objects might be moving
before they collide.
Elastic Collision
When a moving billiard ball makes a headon collision with another billiard ball at
rest, the moving ball comes to rest and the
other ball moves with the speed of the
colliding ball.
We call this an elastic collision; ideally
the colliding objects rebound without
lasting deformation or the generation of
heat.
Inelastic Collision
But momentum is conserved even when the
colliding objects become entangled during the
collision.
This is an inelastic collision, characterized by
deformation or the generation of heat or both. In
a perfectly inelastic collision, both objects stick
together.
Consider, for example, the case of a freight car
moving along a track and colliding with another
freight car at rest.
If the freight cars are of equal mass and are
coupled by the collision, can we predict the
velocity of the coupled cars after impact?
Inelastic Collision
Inelastic collision. The momentum of the
freight car on the left is shared with the
freight car on the right after collision.
Conservation of Momentum
Suppose the single car is moving at 10 meters
per second, and we consider the mass of each
car to be m.
Then, from the conservation of momentum, By
simple algebra, V = 5 m/s.
This makes sense, for since twice as much
mass is moving after the collision, the velocity
must be half as much as the velocity before
collision.
Both sides of the equation are then equal.
Direction is important
Notice the importance of direction in these
cases.
As with any pair of vectors, momenta in
the same direction are simply added.
If two objects are moving toward each
other, one of the momenta is considered
negative, and we combine them by
subtraction.
Inelastic Collision
Inelastic collisions. The net momentum of the
trucks before and after collision is the same.
The same principle applies to gently docking
spacecraft. Their net momentum just before
docking is preserved as their net momentum just
after docking.
Check Yourself
Consider an air track.
Suppose a gliding cart with a mass of 0.5
kg bumps into and sticks to a stationary
cart that has a mass of 1.5 kg.
If the speed of the gliding cart before
impact is v before, how fast will the
coupled carts glide after collision?
Examples
For a numerical example of momentum
conservation, consider a fish that swims
toward and swallows a smaller fish at rest.
If the larger fish has a mass of 5 kg and
swims 1 m/s toward a 1-kg fish, what is
the velocity of the larger fish immediately
after lunch? We will neglect the effects of
water resistance.
Conservation of Momentum
Two fish make up a system, which has the
same momentum just before lunch and
just after lunch.
Complicated Collisions
The net momentum remains unchanged in
any collision, regardless of the angle
between the tracks of the colliding objects.
Expressing the net momentum when
different directions are involved can be
achieved with the parallelogram rule of
vector addition.
We will show some simple examples to
convey the concept.
Momentum is a vector quantity
After the firecracker bursts, the momenta
of its fragments add up (by vector addition)
to the original momentum.
Whatever the nature of a collision or however
complicated it is, the total momentum before, during, and
after remains unchanged.
This extremely useful law enables us to learn much from
collisions without knowing any details about the
interaction forces that act in the collision.
We will see in the next chapter that energy as well as
momentum is conserved.
By applying momentum and energy conservation to the
collisions of subatomic particles as observed in various
detection chambers, we can compute the masses of
these tiny particles.
We obtain this information by measuring momenta and
energy before and after collisions. Remarkably, this
achievement is possible without any exact knowledge of
the forces that act.
Conservation of momentum and
conservation of energy (next chapter) are
the two most powerful tools of mechanics.
Applying them yields detailed information
that ranges from facts about the
interactions of subatomic particles to the
structure and motion of entire galaxies.
animations
Momentum
Quiz 4
1. a. What is the momentum of an 8 Kg
bowling ball rolling at 3 m / s ?
b. If the bowling ball rolls into a pillow and
stops in 0.2 s, calculate the average
force it exerts on the pillow.
2. a. What impulse occurs when an
average force of 15 N is exerted on a
cart for 3 s?
b. If the mass of the cart is 3 Kg and the
cart is initially at rest calculate its final
speed.
Quiz 5
1. A 40 Kg projectile leaves a 2000 Kg
Launcher with a speed of 300m/s.
What is the recoil speed of the launcher?
2. An 9000Kg truck moving at 10 m / s
collides with a 1500 Kg car moving at
20 m / s in the opposite direction. If they
stick together after impact, how fast, and
in what direction, will they be moving?
Momentum
A heavy truck is harder to stop than a
small car moving at the same speed.
We state this fact by saying that the truck
has more momentum than the car.
By momentum -
Force and Momentum
Changes in momentum may occur when there is
either a change in the mass of an object, a
change in velocity, or both.
If momentum changes while the mass remains
unchanged, as is most often the case, then the
velocity changes.
Acceleration occurs.
And what produces an acceleration?
The answer is a force.
The greater the force that acts on an object, the
greater will be the change in velocity and,
hence, the change in momentum.
Time and Momentum
But something else is important in changing
momentum: time—how long a time the force
acts.
Apply a force briefly to a stalled automobile, and
you produce a small change in its momentum.
Apply the same force over an extended period of
time, and a greater change in momentum
results.
A long sustained force produces more change in
momentum than the same force applied briefly.
Force and Time-Impulse
So for changing the momentum of an object,
both force and the time during which the force
acts are important. We name the product of
force and this time interval impulse.
Whenever you exert a net force on something,
you also exert an impulse.
The resulting acceleration depends on the net
force;
the resulting change in momentum depends on
both the net force and the time during which that
force acts.
Check Yourself
1. Which has more momentum, a 1-ton car
moving at 100 km/h or a 2-ton truck
moving at 50 km/h?
2. Does a moving object have impulse?
3. Does a moving object have
momentum?
Impulse Changes Momentum
Impulse changes momentum in much the same
way that force changes velocity.
The relationship of impulse to change of
momentum comes from Newton's second law (a
= F/m).
The time interval of impulse is “buried” in the
term for acceleration (change in velocity/time
interval).
Rearrangement of Newton's second law gives
“force multiplied by the time during which it acts
equals change in momentum.”
Truck into Haystack
A large change in momentum in a long
time requires a small force.
Truck into Wall
A large change in momentum in a short
time requires a large force.
Check Yourself
1. If the boxer is able to make the duration of impact
three times as long by riding with the punch, by how
much will the force of impact be reduced?
2. If the boxer instead moves into the punch such as to
decrease the duration of impact by half, by how much
will the force of impact be increased?
3. A boxer being hit with a punch contrives to extend time
for best results, whereas a karate expert delivers a force
in a short time for best results. Isn't there a contradiction
here?
4. When does impulse equal momentum?
Bouncing
You know that if a flower pot falls from a shelf onto your head, you
may be in trouble.
And whether you know it or not, if it bounces from your head, you
may be in more serious trouble.
Impulses are greater when bouncing takes place.
This is because the impulse required to bring something to a stop
and then, in effect, “throw it back again” is greater than the impulse
required merely to bring something to a stop.
Suppose, for example, that you catch the falling pot with your hands.
Then you provide an impulse to catch it and reduce its momentum to
zero.
If you were to then throw the pot upward, you would have to provide
additional impulse.
So it would take more impulse to catch it and throw it back up than
merely to catch it.
The same greater impulse is supplied by your head if the pot
bounces from it.
Check Yourself
Check Yourself
1. How does the force that the woman
exerts on the bricks compare with the
force exerted on her hand?
2. How will the impulse resulting from the
impact differ if her hand bounces back
upon striking the bricks?
Karate Chop
External Impulse Changes
Momentum
Recall what Newton's second law tells us: If we
want to accelerate an object, we must apply a
force.
We say much the same thing in this chapter
when we say that to change the momentum of
an object we must apply an impulse.
In any case, the force or impulse must be
exerted on the object or any system of objects
by something external to the object or system.
Internal forces don't count.
Internal Forces do not Change
Momentum
For example, the molecular forces within a baseball have
no effect on the momentum of the baseball, just as a
person sitting inside an automobile pushing against the
dashboard, with the dashboard pushing back, has no
effect in changing the momentum of the automobile.
This is because these forces are internal forces, that is,
forces that act and react within the systems themselves.
An outside, or external, force acting on the baseball or
automobile is required for a change in momentum.
If no external force is present, then no change in
momentum is possible.
Conservation of Momentum
When a bullet is fired from a rifle, the forces
present are internal forces.
The total momentum of the system comprising
the bullet and rifle therefore undergoes no net
change.
By Newton's third law of action and reaction, the
force exerted on the bullet is equal and opposite
to the force exerted on the rifle.
The forces acting on the bullet and rifle act for
the same time, resulting in equal but oppositely
directed impulses and therefore equal and
oppositely directed momenta (the plural form of
momentum).
Conservation of Momentum
The recoiling rifle has just as much momentum
as the speeding bullet.
Although both the bullet and rifle by themselves
have gained considerable momentum, the bullet
and rifle together as a system experience no net
change in momentum.
Before firing, the momentum was zero; after
firing, the net momentum is still zero. No
momentum is gained and no momentum is lost
Rifle-Bullet Example
Two important ideas are to be learned from the
rifle-and-bullet example.
The first is that momentum, like velocity, is a
quantity that is described by both magnitude and
direction; we measure both “how much” and
“which direction.”
Momentum is a vector quantity. Therefore,
when momenta act in the same direction, they
are simply added; when they act in opposite
directions, they are subtracted
Rifle-Bullet Example
The second important idea to be taken from the
rifle-and-bullet example is the idea of
conservation.
The momentum before and after firing is the
same.
For the system of rifle and bullet, no momentum
was gained; none was lost.
When a physical quantity remains unchanged
during a process, that quantity is said to be
conserved.
We say momentum is conserved.
Check Yourself
1. A high-speed bus and an innocent bug have a
head-on collision. The sudden change of
momentum for the bug spatters it all over the
windshield. Is the change in momentum of the
bus greater, less, or the same as the change in
momentum experienced by the unfortunate bug?
2. The Starship Enterprise is being pursued in
space by a Klingon warship moving at the same
speed. What happens to the speed of the
Klingon ship when it fires a projectile at the
Enterprise? What happens to the speed of the
Enterprise when it returns the fire?
Collisions
Momentum is conserved in collisions—that is, the net
momentum of a system of colliding objects is unchanged
before, during, and after the collision.
This is because the forces that act during the collision
are internal forces—forces acting and reacting within the
system itself.
There is only a redistribution or sharing of whatever
momentum exists before the collision.
In any collision, we can say
Net momentum before collision = net momentum
after collision.
This is true no matter how the objects might be moving
before they collide.
Elastic Collisions
When a moving billiard ball makes a headon collision with another billiard ball at
rest, the moving ball comes to rest and the
other ball moves with the speed of the
colliding ball.
We call this an elastic collision; ideally
the colliding objects rebound without
lasting deformation or the generation of
heat.
Inelastic Collision
But momentum is conserved even when the colliding
objects become entangled during the collision.
This is an inelastic collision, characterized by
deformation or the generation of heat or both. In a
perfectly inelastic collision, both objects stick together.
Consider, for example, the case of a freight car moving
along a track and colliding with another freight car at
rest.
If the freight cars are of equal mass and are coupled by
the collision, can we predict the velocity of the coupled
cars after impact?
Check Yourself
Consider an air track.
Suppose a gliding cart with a mass of 0.5
kg bumps into and sticks to a stationary
cart that has a mass of 1.5 kg.
If the speed of the gliding cart before
impact is v before, how fast will the
coupled carts glide after collision?
Example
For a numerical example of momentum
conservation, consider a fish that swims
toward and swallows a smaller fish at rest.
If the larger fish has a mass of 5 kg and
swims 1 m/s toward a 1-kg fish, what is
the velocity of the larger fish immediately
after lunch? We will neglect the effects of
water resistance.
Direction is Important
As with any pair of vectors, momenta in
the same direction are simply added.
If two objects are moving toward each
other, one of the momenta is considered
negative, and we combine them by
subtraction.
Complicated Collisions
The net momentum remains unchanged in any
collision, regardless of the angle between the
tracks of the colliding objects.
Expressing the net momentum when different
directions are involved can be achieved with the
parallelogram rule of vector addition.
We will not treat such complicated cases in great
detail here but will show some simple examples
to convey the concept
Whatever the nature of a collision or however
complicated it is, the total momentum before, during, and
after remains unchanged.
This extremely useful law enables us to learn much from
collisions without knowing any details about the
interaction forces that act in the collision.
We will see in the next chapter that energy as well as
momentum is conserved.
By applying momentum and energy conservation to the
collisions of subatomic particles as observed in various
detection chambers, we can compute the masses of
these tiny particles.
We obtain this information by measuring momenta and
energy before and after collisions. Remarkably, this
achievement is possible without any exact knowledge of
the forces that act.
Conservation of momentum and
conservation of energy (next chapter) are
the two most powerful tools of mechanics.
Applying them yields detailed information
that ranges from facts about the
interactions of subatomic particles to the
structure and motion of entire galaxies.