Download Lesson 1: Newton`s First Law of Motion

Newton's First Law of Motion
Newton's First Law
Newton's first law of motion – sometimes referred to as the "law of inertia."
An object at rest tends to stay at rest and an object in motion tends
to stay in motion with the same speed and in the same direction
unless acted upon by an unbalanced force.
There are two parts to this statement – one which predicts the behavior of
stationary objects and the other which predicts the behavior of moving
objects.
The behaviour of all objects can be described by saying that objects tend to
"keep on doing what they're doing" (unless acted upon by an unbalanced
force). If at rest, they will continue in this same state of rest. If in motion with
an eastward velocity of 5 m/s, they will continue in this same state of motion
(5 m/s, East).
There are many applications of Newton's first law of motion. Consider some
of your experiences in an automobile.
Have you ever experienced inertia (resisting changes in your state of motion)
in an automobile while it is braking to a stop? The force of the road on the
locked wheels provides the unbalanced force to change the car's state of
motion, yet there is no unbalanced force to change your own state of motion.
Thus, you continue in motion, sliding forward along the seat
Yes, seat belts are used to provide safety for passengers whose motion is
governed by Newton's laws. The seat belt provides the unbalanced force
which brings you from a state of motion to a state of rest. Perhaps you could
speculate what would occur when no seat belt is used.
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Inertia and Mass
This is the natural tendency of objects to resist changes in their state of
motion. This tendency to resist changes in their state of motion is described
as inertia.
Inertia is the resistance an object has to a change in its state of motion.
Galileo, the premier scientist of the seventeenth century, developed the
concept of inertia. Galileo reasoned that moving objects eventually stop
because of a force called friction.
Isaac Newton built on Galileo's thoughts about motion. Newton's first law of
motion declares that a force is not needed to keep an object in motion. Slide
a book across a table and watch it slide to a stop. The book in motion on the
table top does not come to rest because of the absence of a force; rather it is
the presence of a force – the force of friction – which brings the book to a
halt. In the absence of a frictional force, the book would continue in motion
with the same speed and in the same direction – forever!
All objects resist changes in their state of motion. All objects have this
tendency – they have inertia. But do some objects have more of a tendency
to resist changes than others? Yes, absolutely! The tendency of an object to
resist changes in its state of motion is dependent upon its mass. Inertia is a
quantity which is solely dependent upon mass.
The more mass an object has, the more inertia it has – the more
tendency it has to resist changes in its state of motion.
State of Motion
Inertia is the tendency of an object to resist changes in its state of motion.
But what does the phrase "state of motion" mean? The state of motion of an
object is defined by its velocity – its speed in a given direction. Thus, inertia
could be redefined as follows:
Inertia is the tendency of an object to resist changes in its velocity.
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An object at rest has zero velocity and (in the absence of an unbalanced
force) will remain at a zero velocity; it will not change its state of motion (i.e.,
its velocity). An object in motion with a velocity of 2 m/s, East (in the absence
of an unbalanced force) will remain in motion with a velocity of 2 m/s, East; it
will not change its state of motion (i.e., its velocity). Objects resist changes in
their velocity.
An object which is not changing its velocity is said to have an acceleration of
0 m/s2. Thus, an alternate definition of inertia would be:
Inertia is the tendency of an object to resist accelerations.
Balanced and Unbalanced Forces
What is an unbalanced force?
Consider an example of a balanced force – a person standing upon the
ground. There are two forces acting upon the person. The force of gravity
exerts a downward force. The push of the floor exerts an upward force.
Since these two forces are of equal magnitude and in opposite directions,
they balance each other. The person is at equilibrium. There is no unbalanced
force acting upon the person and thus the person maintains his/her state of
motion.
A net force (i.e., an unbalanced force) causes an acceleration.
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Newton's Second Law of Motion
Newton's Second Law
Newton's first law of motion predicts the behavior of objects
for which all existing forces are balanced.
Objects at equilibrium (the condition in which all forces
balance) will not accelerate. According to Newton, an object
will only accelerate if there is a net or unbalanced force acting
upon it. The presence of an unbalanced force will accelerate an
object – changing its speed, its direction, or both its speed and direction.
Newton's second law of motion pertains to the behavior of objects for which
all existing forces are not balanced. The second law states that the
acceleration of an object is dependent upon two variables – the net force
acting upon the object and the mass of the object. The acceleration of
an object depends directly upon the net force acting upon the object, and
inversely upon the mass of the object. As the net force increases, so will the
object's acceleration. However, as the mass of the object increases, its
acceleration will decrease.
Newton's second law of motion can be formally stated as follows:
The acceleration of an object as produced by a net force is directly
proportional to the magnitude of the net force, in the same
direction as the net force, and inversely proportional to the mass of
the object.
In terms of an equation, the net force is equal to the product of the object's
mass and its acceleration.
F=m*a
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Thus, the definition of the standard metric unit of
force is given by the above equation.
One Newton is defined as the amount of force required to give a 1kg mass an acceleration of 1 m/s2.
In conclusion, Newton's second law explains the behavior of objects
upon which unbalanced forces are acting. The law states that
unbalanced forces cause objects to accelerate with an acceleration
that is directly proportional to the net force and inversely
proportional to the mass of the object.
The Big Misconception
While most people know what Newton's laws say, many people do not know
what they mean (or simply do not believe what they mean).
Newton's laws declare loudly that a net force (an unbalanced force) causes an
acceleration and the acceleration is in the same direction as the net force. To
test your own belief system, consider the following question and its answer.
Two students are discussing their physics homework prior to class. They are
discussing an object which is being acted upon by two individual forces (both
in a vertical direction); the free-body diagram for the particular object is
shown above.
During the discussion, Anna suggests to David that the object under
discussion could be moving. In fact, Anna suggests that if friction and air
resistance are ignored (because of their negligible magnitudes), the object
could be moving in a horizontal direction. According to Anna, an object
experiencing forces as described by the free-body diagram at the right could
be experiencing a horizontal motion as described by the ticker tape
representation below.
David objects, arguing that the object could not have any horizontal motion if
there are only vertical forces acting upon it. Noah claims that the object must
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be at rest, perhaps on a table or floor. After all, says David, an object
experiencing a balance of forces will be at rest. With whom do you agree?
Imagine for the moment that there was no friction along the level surface
from point B to point C and that there was no air resistance to impede your
motion. How far would your sled travel? And what would its motion be like?
Most students quickly answer: the sled would travel forever at constant
speed.
Without friction or air resistance to slow it down, the sled would continue in
motion with the same speed and in the same direction. The forces acting
upon the sled from point B to point C would be the normal force (the snow
pushing up on the sled) and the gravity force — see diagram above. These
forces are balanced and since the sled is already in motion at point B, it will
continue in motion with the same speed and in the same direction. So, as in
the case of the sled and as in the case of the object which David and Anna
are discussing, an object can be moving to the right even if the only forces
acting upon it are vertical forces.
Forces do not cause motion; forces cause accelerations.
A force is not required to keep a moving book in motion; a force is
not required to keep a moving sled in motion; and a force is not
required to keep any horizontally moving object in motion. Forces do
not cause motion; forces cause accelerations.
Finding Acceleration
The net force is the sum of all the individual forces.
The three major equations which will be useful are:
1. the equation for net force
( F = m * a),
2. the equation for gravitational force ( W = m * g), and
3. the equation for frictional force
( Ffrict = µ * Fnorm).
The process of determining the acceleration of an object demands that the
mass and the net force are known. If mass (m) and net force (Fnet) are
known, then the acceleration is determined by the equation:
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Thus, the task involves using the above equations, the given information, and
your understanding of Newton's laws to determine the acceleration.
Finding Individual Forces
The process of determining the value of the individual forces acting upon an
object involves an application of Newton's second law and an application of
the meaning of the net force. If mass (m) and acceleration (a) are known,
then the net force (F) can be determined by use of the equation:
F =m*a
If the numerical value of the net force and its direction are known, then the
value of all individual forces can be determined. The task involves using the
above equations, the given information, and your understanding of net force
to determine the value of the individual forces.
Free Fall and Air Resistance
All objects (regardless of their mass) free-fall with the same acceleration –
10 m/s2. This acceleration value is so important in physics that it has its own
peculiar name – the acceleration of gravity – and its own peculiar symbol –
"g."
But why do all objects free-fall at the same rate of acceleration regardless of
their mass?
Is it because they all weigh the same?
... because they all have the same gravity?
... because the air resistance is the same for each?
Why?
Why do objects which encounter air resistance ultimately reach a terminal
velocity?
In situations in which there is air resistance, why do massive objects fall
faster than less massive objects?
To answer the above questions, Newton's second law of motion (Fnet = m*a)
will be applied to analyze the motion of objects which are falling under the
influence of gravity only (free-fall) and under the dual influence of gravity and
air resistance.
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Free Fall Motion
Free-fall is a special type of motion in which the only force acting upon an
object is gravity. Objects, which are said to be undergoing free-fall, do not
encounter a significant force of air resistance; they are falling under the sole
influence of gravity. Under such conditions, all objects will fall with the
same rate of acceleration, regardless of their mass. Why? Consider the
free-falling motion of a 10-kg rock and a 1-kg rock.
If Newton's second law were applied to their falling motion, and if free-body
diagrams were constructed, you would see that the 10-kg rock experiences a
greater force of gravity. This greater force of gravity would have a direct
effect upon the rock's acceleration; thus, based on force alone, you might
think that the 10-kg rock would accelerate faster. But acceleration depends
upon two factors: force and mass. The 10-kg rock obviously has more mass
(or inertia) than the 1-kg rock. This increased mass has an inverse effect
upon the rock's acceleration. Thus, the direct effect of greater force on the
10-kg rock is offset by the inverse effect of its greater mass; and so each rock
accelerates at the same rate – 10 m/s2. The ratio of force to mass (F/m) is
the same for each rock in situations involving free fall; this ratio (F/m) is
equivalent to the acceleration of the object.
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Falling with Air Resistance
As an object falls through air, it usually encounters some degree of air
resistance. Air resistance is the result of collisions of the object's
leading surface with air molecules. The actual amount of air resistance
encountered by an object depends upon a variety of factors. The two most
common factors which have a direct effect upon the amount of air resistance
present are the speed of the object and the cross-sectional area of
the object.
Increased speeds result in an increased
amount of air resistance.
Increased cross-sectional areas result in an
increased amount of air resistance.
Terminal Velocity
As an object falls, it picks up speed. This increase in speed leads to
an increase in the amount of air resistance. Eventually, the force of
air resistance becomes large enough to balance the force of gravity.
At this instant in time, the net force is 0 Newtons — the object stops
accelerating. The object is said to have "reached a terminal
velocity."
Any change in velocity terminates as a result of the balancing of the
individual forces acting upon the object. The velocity at which this occurs is
called the "terminal velocity."
In situations in which there is air resistance, massive objects fall faster than
less massive objects. Why?
As you learned above, the amount of air resistance depends upon the speed
of the object. Objects like the skydivers above will continue to accelerate to
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higher speeds until they encounter an amount of air resistance which is equal
to their weight. Since the 150-kg skydiver weighs more (experiences a greater
force of gravity), he will have to accelerate to a higher speed before reaching
his terminal velocity. Thus, massive objects fall faster than less massive
objects because they are acted upon by a larger force of gravity; for this
reason, they accelerate to higher speeds until the air resistance force equals
their gravity force.
Newton's Third Law of Motion
Newton's Third Law
A force is a push or a pull upon an object which
results from its interaction with another object. Forces result from
interactions! Some forces result from contact interactions (normal,
frictional, tensional, and applied forces are examples of contact
forces) and other forces result from action-at-a-distance
interactions (gravitational, electrical, and magnetic forces are
examples of action-at-a-distance forces).
Formally stated, Newton's third law is:
"For every action, there is an equal and opposite reaction."
The statement means that in every interaction, there is a pair of forces acting
on the two interacting objects. The size of the force on the first object equals
the size of the force on the second object. The direction of the force on the
first object is opposite to the direction of the force on the second object.
Forces always come in pairs – equal and opposite action-reaction
force pairs.
A variety of action-reaction force pairs are evident in nature. Consider the
propulsion of a fish through the water. A fish uses its fins to push water
backwards. But a push on the water will only serve to accelerate the water. In
turn, the water reacts by pushing the fish forwards, propelling the fish
through the water. The size of the force on the water equals the size of the
force on the fish; the direction of the force on the water (backwards) is
opposite to the direction of the force on the fish (forwards). For every action,
there is an equal (in size) and opposite (in direction) reaction force. Actionreaction force pairs make it possible for fishes to swim.
Consider the flying motion of birds. A bird flies by use of its wings. The wings
of a bird push air downwards. In turn, the air reacts by pushing the bird
upwards. The size of the force on the air equals the size of the force on the
bird; the direction of the force on the air (downwards) is opposite to the
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direction of the force on the bird (upwards). For every action, there is an
equal (in size) and opposite (in direction) reaction. Action-reaction force pairs
make it possible for birds to fly.
Consider the motion of your automobile on your way to school. An automobile
is equipped with wheels that spin backwards. As the wheels spin backwards,
they push the road backwards. In turn, the road reacts by pushing the wheels
forward. The size of the force on the road equals the size of the force on the
wheels (or automobile); the direction of the force on the road (backwards) is
opposite to the direction of the force on the wheels (forwards). For every
action, there is an equal (in size) and opposite (in direction) reaction. Actionreaction force pairs make it possible for automobiles to move.
Newton's Third Law of Motion
Identifying Action and Reaction Force Pairs
Identifying and describing action-reaction force pairs is a simple matter of
identifying the two interacting objects and making two statements describing
who is pushing on whom and in which direction. For example, consider the
interaction between a baseball bat and a baseball.
The baseball forces the bat to the right (an action); the bat forces the ball to
the left (the reaction). Note that the nouns in the sentence describing the
action force switch places when describing the reaction force.
Consider the following three examples. The action force is stated; determine
the reaction force.
Athlete pushes bar upwards.
Bowling ball pushes pin rightwards.
Compressed air pushes balloon wall outwards.
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Momentum
Momentum
Momentum is a physics term; it refers to the quantity of motion that
an object has. Momentum can be defined as "mass in motion."
All objects have mass; so if an object is moving, then it has momentum - it
has its mass in motion. The amount of momentum which an object has is
dependent upon two variables: how much stuff is moving and how
fast the stuff is moving. Momentum depends upon the variables mass
and velocity. In terms of an equation, the momentum of an object is equal
to the mass of the object times the velocity of the object.
Momentum = mass * velocity
Momentum = m * v
where m = mass and v=velocity. The equation illustrates that momentum is
directly proportional to an object's mass and directly proportional to the
object's velocity.
The units for momentum would be mass units times velocity units. The
standard metric unit of momentum is the kg*m/s.
Momentum is a vector quantity. As discussed in an
earlier unit, a vector quantity is a quantity which is fully
described by both magnitude and direction.
From the definition of
momentum, it becomes obvious
that an object has a large momentum if either its mass
or its velocity is large. Both variables are of equal
importance in determining the momentum of an
object.
Consider a Mack truck and a roller skate moving down the street at the same
speed. The considerably greater mass of the Mack truck gives it a
considerably greater momentum. Yet if the Mack truck were at rest, then the
momentum of the least massive roller skate would be the greatest; for the
momentum of any object which is at rest is 0.
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Objects at rest do not have momentum - they do not have any "mass
in motion." Both variables - mass and velocity - are important in
comparing the momentum of two objects.
Momentum and Impulse Connection
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.
In rugby, 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.
These concepts are merely an outgrowth of Newton's second law . Newton's
second law (F=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.
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If both sides of the above equation are multiplied by the quantity t, a new
equation results.
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
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
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.



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
Real-World Applications
In a collision, an object experiences a force for a given amount of time which
results in its mass undergoing a change in velocity (i.e., which results in a
momentum change).
There are four physical quantities mentioned in the above statement - force,
time, mass, and velocity change.
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The force multiplied by the time is known as the impulse and the
mass multiplied by the velocity change is known as the change in
momentum. The impulse experienced by an object is always equal to the
change in its momentum. In terms of equations, this was expressed as
This is known as the impulse-momentum change theorem.
In particular, we will focus upon


the effect of collision time upon the amount
of force an object experiences, and
the effect of rebounding upon the velocity
change and hence the amount of force an
object experiences.
The Effect of Collision Time upon the Force
The greater the time over which the collision occurs, the smaller the force
acting upon the object. Thus, to minimize the effect of the force on an object
involved in a collision, the time must be increased; and to maximize the effect
of the force on an object involved in a collision, the time must be decreased.

There are several real-world applications of this phenomena. One example
is the use of air bags in automobiles. Air bags are used in automobiles
because they are able to minimize the effect of the force on an object
involved in a collision. Air bags accomplish this by extending the time
required to stop the momentum of the driver and passenger. When
encountering a car collision, the driver and passenger tend to keep moving
in accord with Newton's first law. Their motion carries them towards a
windshield which results in a large force exerted over a short time in order
to stop their momentum. If instead of hitting the windshield, the driver
and passenger hit an air bag, then the time duration of the impact is
increased. When hitting an object with some give such as an air bag, the
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time duration might be increased by a factor of 100. Increasing the time
by a factor of 100 will result in a decrease in force by a factor of 100. Now
that's physics in action.
The same principle explains why dashboards are padded. If the air bags
do not deploy (or are not installed in a car), then the driver and passengers
run the risk of stopping their momentum by means of a collision with the
windshield or the dashboard. If the driver or passenger should hit the
dashboard, then the force and time required to stop their momentum is
exerted by the dashboard. Padded dashboards provide some give in such a
collision and serve to extend the time duration of the impact, thus minimizing
the effect of the force. This same principle of padding a potential impact area
can be observed in gymnasiums (underneath the basketball hoops), in polevaulting pits, in baseball gloves and goalie mitts, on the fist of a boxer, inside
the helmet of a football player, and on gymnastic mats.
Fans of boxing frequently observe this same principle
of minimizing the effect of a force by extending the
time of collision. When a boxer recognizes that he
will be hit in the head by his opponent, the boxer
often relaxes his neck and allows his head to move
backwards upon impact. In the boxing world, this is
known as riding the punch. A boxer rides the punch
in order to extend the time of impact of the glove
with their head. Extending the time results in decreasing the force and thus
minimizing the effect of the force in the collision. Merely increasing the
collision time by a factor of ten would result in a tenfold decrease in the force.
Now that's physics in action.
Nylon ropes are used in the sport of rock-climbing for the same reason. Rock
climbers attach themselves to the steep cliffs by means of
nylon ropes. If a rock climber should lose her grip on the
rock, she will begin to fall. In such a situation, her
momentum will ultimately be halted by means of the rope,
thus preventing a disastrous fall to the ground below. The
ropes are made of nylon or similar material because of its
ability to stretch. If the rope is capable of stretching upon
being pulled taut by the falling climber's mass, then it will
apply a force upon the climber over a longer time period.
Extending the time over which the climber's momentum is broken results in
reducing the force exerted on the falling climber. For certain, the rock climber
can appreciate minimizing the effect of the force through the use of a longer
time of impact. Now that's physics in action.
In racket and bat sports, hitters are often encouraged
to follow-through when striking a ball. High speed films
of the collisions between bats/rackets and balls have
shown that the act of following through serves to
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increase the time over which a collision occurs. This increase in time must
result in a change in some other variable in the impulse-momentum change
theorem. Surprisingly, the variable which is dependent upon the time in such
a situation is not the force. The force in hitting is dependent upon how hard
the hitter swings the bat or racket, not the time of impact. Instead, the
follow-through increases the time of collision and subsequently contributes to
an increase in the velocity change of the ball. By following through, a hitter
can hit the ball in such a way that it leaves the bat or racket with more
velocity (i.e., the ball is moving faster). In tennis, baseball, racket ball, etc.,
giving the ball a high velocity often leads to greater success.
The Effect of Rebounding
Occasionally when objects collide, they bounce off each other (as opposed to
sticking to each other and traveling with the same speed after the collision).
Bouncing off each other is known as rebounding. Rebounding involves a
change in direction of an object; the before- and after-collision direction is
different. Rebounding situations are characterized by a large velocity change
and a large momentum change.
From the impulse-momentum change theorem, we could deduce that a
rebounding situation must also be accompanied by a large impulse. Since the
impulse experienced by an object equals the momentum change of the
object, a collision characterized by a large momentum change must also be
characterized by a large impulse.
The importance of rebounding is critical to the outcome of automobile
accidents. In an automobile accident, two cars can either collide and bounce
off each other or collide and crumple together and travel together with the
same speed after the collision. But which would be more damaging to the
occupants of the automobiles - the rebounding of the cars or the crumpling
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up of the cars? Contrary to popular opinion, the crumpling up of cars is the
safest type of automobile collision. As mentioned above, if cars rebound upon
collision, the momentum change will be larger and so will the impulse. A
greater impulse will typically be associated with a bigger force. Occupants of
automobiles would certainly prefer small forces upon their bodies during
collisions. In fact, automobile designers and safety engineers have found
ways to reduce the harm done to occupants of automobiles by designing cars
which crumple upon impact. Automobiles are made with crumple zones.
Crumple zones are sections in cars which are designed to crumple up when
the car encounters a collision. Crumple zones minimize the effect of the force
in an automobile collision in two ways. By crumpling, the car is less likely to
rebound upon impact, thus minimizing the momentum change and the
impulse. Finally, the crumpling of the car lengthens the time over which the
car's momentum is changed; by increasing the time of the collision, the force
of the collision is greatly reduced. total momentum of the two objects before
the collision is equal to the total momentum of the two objects after the
collision. That is, the momentum lost by object 1 is equal to the momentum
gained by object 2.
The above statement tells us that the total momentum of a collection of
objects (a system) is conserved" - that is the total amount of momentum is a
constant or unchanging value.
Consider a collision between two objects - object 1 and object 2. For such a
collision, the forces acting between the two objects are equal in magnitude
and opposite in direction (Newton's third law). This statement can be
expressed in equation form as follows.
The forces act between the two objects for a given amount of time. In some
cases, the time is long; in other cases the time is short. Regardless of how
long the time is, it can be said that the time that the force acts upon object 1
is equal to the time that the force acts upon object 2. This is merely logical;
forces result from interactions (or touching) between two objects. If object 1
touches object 2 for 0.050 seconds, then object 2 must be touching object 1
for the same amount of time (0.050 seconds). As an equation, this can be
stated as
Since the forces between the two objects are equal in magnitude and
opposite in direction, and since the times for which these forces act are equal
in magnitude, it follows that the impulses experienced by the two objects are
also equal in magnitude and opposite in direction. As an equation, this can be
stated as
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But the impulse experienced by an object is equal to the change in
momentum of that object (the impusle-momentum change theorem). Thus,
since each object experiences equal and opposite impulses, it follows logically
that they must also experience equal and opposite momentum changes. As
an equation, this can be stated as
The above equation is one statement of the law of momentum conservation.
In a collision, the momentum change of object 1 is equal and opposite to the
momentum change of object 2. That is, the momentum lost by object 1 is
equal to the momentum gained by object 2. In a collision between two
objects, one object slows down and loses momentum while the other object
speeds up and gains momentum. If object 1 loses 75 units of momentum,
then object 2 gains 75 units of momentum. Yet, the total momentum of the
two objects (object 1 plus object 2) is the same before the collision as it is
after the collision; the total momentum of the system (the collection of two
objects) is conserved.
The Law of Momentum Conservation
A useful means of depicting the transfer and the conservation of money
between Jack and Jill is by means of a table.
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The table shows the amount of money possessed by the two individuals
before and after the interaction. It also shows the total amount of money
before and after the interaction. Note that the total amount of money ($200)
is the same before and after the interaction - it is conserved. Finally, the table
shows the change in the amount of money possessed by the two individuals.
Note that the change in Jack's money account (-$50) is equal and opposite to
the change in Jill's money account (+$50).
For any collision occurring in an isolated system, momentum is conserved the total amount of momentum of the collection of objects in the system is
the same before the collision as after the collision. This is the very
phenomenon which was observed in "The Cart and The Brick" lab. In this lab,
a brick at rest was dropped upon a loaded cart which was in motion.
Before the collision, the dropped brick had 0 units of momentum (it was at
rest). The momentum of the loaded cart can be determined using the velocity
(as determined by the ticker tape analysis) and the mass. The total amount of
momentum was the sum of the dropped brick's momentum (0 units) and the
loaded cart's momentum. After the collision, the momenta of the two
separate objects (dropped brick and loaded cart) can be determined from
their measured mass and their velocity (found from the ticker tape analysis).
If momentum is conserved during the collision, then the sum of the dropped
brick's and loaded cart's momentum after the collision should be the same as
before the collision. The momentum lost by the loaded cart should equal (or
approximately equal) the momentum gained by the dropped brick.
Momentum data for the interaction between the dropped brick and the loaded
cart could be depicted in a table similar to the money table above.
Before
Collision
Momentum
After
Collision
Momentum
Change in
Momentum
Dropped Brick
0 units
14 units
+14 units
Loaded Cart
45 units
31 units
-14 units
Total
45 units
45 units
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Note that the loaded cart lost 14 units of momentum and the dropped brick
gained 14 units of momentum. Note also that the total momentum of the
system (45 units) was the same before the collision as it is after the collision.
Collisions commonly occur in contact sports (such as rugby) and racket and
bat sports (such as baseball, golf, tennis, etc.). Consider a collision in rugby
between a fullback and a linebacker during a goal-line stand. The fullback
plunges across the goal line and collides in midair with linebacker. The
linebacker and fullback hold each other and travel together after the collision.
The fullback possesses a momentum of 100 kg*m/s, East before the collision
and the linebacker possesses a momentum of 120 kg*m/s, West before the
collision. The total momentum of the system before the collision is 20 kg*m/s,
West . Therefore, the total momentum of the system after the collision must
also be 20 kg*m/s, West. The fullback and the linebacker move together as a
single unit after the collision with a combined momentum of 20 kg*m/s.
Now suppose that a medicine ball is thrown to a clown who is at rest upon
the ice; the clown catches the medicine ball and glides together with the ball
across the ice. The momentum of the medicine ball is 80 kg*m/s before the
collision. The momentum of the clown is 0 m/s before the collision. The total
momentum of the system before the collision is 80 kg*m/s. Therefore, the
total momentum of the system after the collision must also be 80 kg*m/s.
The clown and the medicine ball move together as a single unit after the
collision with a combined momentum of 80 kg*m/s. Momentum is conserved
in the collision.
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Isolated Systems
For a collision occurring between object 1 and
object 2 in an isolated system, the total
momentum of the two objects before the
collision is equal to the total momentum of the
two objects after the collision. That is, the
momentum lost by object 1 is equal to the
momentum gained by object 2.
Total system momentum is conserved for
collisions occurring in isolated systems. But what
makes a system of objects an isolated system?
And is momentum conserved if the system is not
isolated?
A system is a collection of two or more objects. An isolated system is a
system which is free from the influence of a net external force.
Consider the collision of two balls on the billiards table. The collision occurs in
an isolated system as long as friction is small enough that its influence upon
the momentum of the billiard balls can be neglected. If so, then the only
unbalanced forces acting upon the two balls are the contact forces which they
apply to one another. These two forces are considered internal forces since
they result from a source within the system - that source being the contact of
the two balls. For such a collision, total system momentum is conserved.
If a system is not isolated, then the total system momentum is not
conserved.
Suppose Jack and Jill (each with $100 in their pockets) undergo a financial
interaction in which Jack hands Jill $50 for the purchase of some goods. If
Jack and Jill were isolated from the influence of the rest of the world, then
Jack would end up with $50 and Jill would end up with $150. The total money
in the system would be $200 both before and after the transaction; total
system money would be conserved. If however, a third influence enters from
outside of the system to take away or (more fortunately) to add money to the
system,( V.A.T.) then total system momentum would not be conserved. If a
thief interfered with his filthy hands so as to steal $20, then perhaps Jack
would finish with $40 and Jill would finish with $140. In the case of a non-
isolated system, the total momentum is not conserved.
Elastic collisions are ones in which two objects collide and then
move apart having lost little or no momentum.
Inelastic collisions are ones in which two objects collide and then
stick together and move together after the collision.
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The Law of Momentum Conservation
Consider the following problem:
A 15-kg medicine ball is thrown at a velocity of 20 km/hr to a 60-kg person
who is at rest on ice. The person catches the ball and subsequently slides
with the ball across the ice. Determine the velocity of the person and the ball
after the collision.
Such a motion can be considered as a collision between a person and a
medicine ball. Before the collision, the ball has momentum and the person
does not. The collision causes the ball to lose momentum and the person to
gain momentum. After the collision, the ball and the person travel with the
same velocity ("v") across the ice.
If it can be assumed that the effect of friction between the person and the ice
is negligible, then the collision is elastic and has occurred in an isolated
system. Momentum should be conserved and the problem can be solved for v
by use of a momentum table as shown below.
Before Collision
After Collision
Person
0
60 * v
Medicine ball
300
15 * v
Total
300
300
Now consider a similar problem involving momentum conservation.
Granny (m=80 kg) whizzes around the rink with a velocity of 6 m/s. She
suddenly collides with Ambrose (m=40 kg) who is at rest directly in her path.
Rather than knock him over, she picks him up and continues in motion
without "braking." Determine the velocity of Granny and Ambrose. Assume
that no external forces act on the system so that it is an isolated system.
Before the collision, Granny has momentum and Ambrose does not. The
collision causes Granny to lose momentum and Ambrose to gain momentum.
After the collision, the Granny and Ambrose move with the same velocity ("v")
across the rink.
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Since the collision between Granny and Ambrose occurs in an isolated system,
total system momentum is conserved. The total momentum before the
collision (possessed solely by Granny) equals the total momentum after the
collision (shared between Granny and Ambrose). The table below depicts this
principle of momentum conservation.
Before Collision
After Collision
Granny
80 * 6 = 480
80 * v
Ambrose
0
40 * v
Total
480
480
Observe in the table above that the known information about the mass and
velocity of Granny and Ambrose was used to determine the before-collision
momenta of the individual objects and the total momentum of the system.
Since momentum is conserved, the total momentum after the collision is
equal to the total momentum before the collision
The two collisions above are examples of inelastic collisions.
Technically, an inelastic collision is a collision in which the kinetic energy of
the system of objects is not conserved. In an inelastic collision, the kinetic
energy of the colliding objects is transformed into other non-mechanical
forms of energy such as heat energy and sound energy. To simplify matters,
we will consider any collisions in which the two colliding objects stick together
and move with the same post-collision speed to be an extreme example of an
inelastic collision.
Now we will consider the analysis of a collision in which the two objects do
not stick together. In this collision, the two objects will bounce off each other.
While this is not technically an elastic collision, it is more elastic than collisions
in which the two objects stick together.
A 3000-kg truck moving with a velocity of 10 m/s hits a 1000-kg parked car.
The impact causes the 1000-kg car to be set in motion at 15 m/s. Assuming
that momentum is conserved during the collision, determine the velocity of
the truck after the collision. In this collision, the truck has a considerable
amount of momentum before the collision and the car has no momentum (it
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is at rest). After the collision, the truck slows down (loses momentum) and
the car speeds up (gains momentum).
The collision can be analyzed using a momentum table similar to the above
situations.
Before Collision
After Collision
Truck
3000 * 10 = 30 000
3000 * v
Car
0
1000 * 15 = 15 000
Total
30 000
30 000
Using Equations as a Guide to Thinking
The process of solving this problem involved using a conceptual
understanding of the equation for momentum (p=m*v). This equation
becomes a guide to thinking about how a change in one variable effects a
change in another variable. The constant quantity in a collision is the
momentum (momentum is conserved). For a constant momentum value,
mass and velocity are inversely proportional. Thus, an increase in mass
results in a decrease in velocity.
A large fish is in motion at 2 m/s when it encounters a smaller fish which is at
rest. The large fish swallows the smaller fish and continues in motion at a
reduced speed. If the large fish has three times the mass of the smaller fish,
then what is the speed of the large fish (and the smaller fish) after the
collision?
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A railroad diesel engine has four times the mass of a flatcar. If a diesel coasts
at 5 km/hr into a flatcar that is initially at rest, how fast do the two coast if
they couple together?
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