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Momentum and Collisions
What is momentum?
• A vector quantity defined as the product of an objects mass and
velocity
• Momentum can be defined as "mass in motion."
• All objects have mass; so if an object is moving, then it has
momentum
• Mass in motion.
Momentum Formula
• P=mv
• Momentum= mass x velocity
• Momentum is directly proportional to an object's mass and directly
proportional to the object's velocity.
Momentum has direction
• The direction of the momentum vector is the same as the
direction of the velocity of the object.
• The direction of the velocity vector is the same as the direction
that an object is moving.
• As a vector quantity, the momentum of an object is fully described
by both magnitude and direction.
• Ex: 5 kg ball moving eastward
Double the mass= Double the momentum
• The momentum equation can help us to think about how a change
in one of the two variables might affect the momentum of an
object. Consider a 0.5-kg physics cart loaded with one 0.5-kg brick
and moving with a speed of 2.0 m/s. The total mass of loaded cart
is 1.0 kg and its momentum is 2.0 kg•m/s. If the cart was instead
loaded with three 0.5-kg bricks, then the total mass of the loaded
cart would be 2.0 kg and its momentum would be 4.0 kg•m/s. A
doubling of the mass results in a doubling of the momentum.
Ex: problem
• A 2250 kg pickup truck has a velocity of 25 m/s to the east. What
is the momentum of the truck?
Answer
• 5.6 x 10^4 kg * m/s to the east
Questions
• 2. A car possesses 20 000 units of momentum. What would be the
car's new momentum if ...
• a. its velocity was doubled.
• b. its velocity was tripled.
• c. its mass was doubled (by adding more passengers and a greater
load)
• d. both its velocity was doubled and its mass was doubled.
Answer
• A. p = 40 000 units (doubling the velocity will double the
momentum)
• B. p = 60 000 units (tripling the velocity will triple the momentum)
• C. p = 40 000 units (doubling the mass will double the momentum)
• D. p = 80 000 units (doubling the velocity will double the
momentum and doubling the mass will also double the
momentum; the combined result is that the momentum is doubled
twice -quadrupled)
Concept of momentum
• Any 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 that an object has, the harder that it is to
stop.
• A greater amount of force or a longer amount of time or both to
bring such an object 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.
Velocity of an object
• A force acting for a given amount of time will change an object's
momentum.
• 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.
Impulse
• For a constant external force, the product of the force and the
time over which it acts on an object.
Newton’s second law
• 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.
• F=m•a
• or
• F = m • ∆v / t
• If newtons equations are multiplied by the quantity t, a new
equation results.
• F • t = m • ∆v
• This equation represents one of two primary principles to be used
in the analysis of collisions during 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.
Impulse
• In physics, the quantity Force • time is known as impulse. And
since the quantity m•v is the momentum, the quantity m•Δv must
be the change in momentum. The equation really says that the
• Impulse = Change in momentum
• The physics of collisions are governed by the laws of momentum;
Rebounding Collisions
• Now consider a collision of a tennis ball with a wall. Depending on
the physical properties of the ball and wall, the speed at which
the ball rebounds from the wall upon colliding with it will vary.
• 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 in addition to a speed
change. The result of the direction change is a 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.
Elastic collision
• In general, elastic collisions are characterized by a large velocity
change, a large momentum change, a large impulse, and a large
force.
Impulse-momentum change equation and law of
momentum
• In a collision, an object experiences a force for a specific amount
of time that results in a change in momentum. The result of the
force acting for the given amount of time is that the object's mass
either speeds up or slows down (or changes direction). The
impulse experienced by the object equals the change in
momentum of the object.
• In equation form, F • t = m • Δ v.
Collision
• 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 a time equivalent to approximately nine dots. In the halfbackdefensive back collision, the halfback experiences a force that 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.
Force and Impulse
• A 1400 Kg car moving westward with a velocity of 15 m/s collides
with a utility pole and is brought to rest in 0.30s. Find the
magnitude of the force excerted on the car during the collision.
Answer
• F ∆ 𝑡 = ∆𝑝 = 𝑚𝑣𝑓 − 𝑚𝑣𝑖
• F= mvf- mvi / ∆ 𝑡
• F= 7.0 X 10 ^4 N to the east
Greatest velocity change?
Greatest acceleration?
Greatest momentum change?
Greatest Impulse?
• a. The velocity change is greatest in case A. The v changes from +5 m/s
to -3 m/s. This is a change of 8 m/s (-) and is greater than in case B (-4
m/s).
• b. The acceleration is greatest in case A. Acceleration depends on
velocity change and the velocity change is greatest in case A (as stated
above).
• c. The momentum change is greatest in case A. Momentum change
depends on velocity change and the velocity change is greatest in case A
(as stated above).
• d. The impulse is greatest in case A. Impulse equals momentum change
and the momentum change is greatest in case A (as stated above).
Impulse-momentum
• 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.
CK-12 Interactive
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Newton’s third law in Collision’s
• Newton's third law of motion is naturally applied to collisions
between two objects. In a collision between two objects, both
objects experience forces that are equal in magnitude and
opposite in direction. Such forces often cause one object to speed
up (gain momentum) and the other object to slow down (lose
momentum).
• In accord with Newton's second law of motion, the acceleration of
an object is dependent upon both force and mass. Thus, if the
colliding objects have unequal mass, they will have unequal
accelerations as a result of the contact force that results during
the collision.
• Consider the collision between the club head and the golf ball in
the sport of golf. When the club head of a moving golf club
collides with a golf ball at rest upon a tee, the force experienced
by the club head is equal to the force experienced by the golf
ball. Most observers of this collision have difficulty with this
concept because they perceive the high speed given to the ball as
the result of the collision. They are not observing unequal forces
upon the ball and club head, but rather unequal accelerations.
Both club head and ball experience equal forces, yet the ball
experiences a greater acceleration due to its smaller mass.
• In a collision, there is a force on both objects that causes an
acceleration of both objects. The forces are equal in magnitude
and opposite in direction, yet the least massive object receives
the greatest acceleration
• Consider the interaction between a male and female figure skater
in pair figure skating. A woman (m = 45 kg) is kneeling on the
shoulders of a man (m = 70 kg); the pair is moving along the ice at
1.5 m/s. The man gracefully tosses the woman forward through
the air and onto the ice. The woman receives the forward force
and the man receives a backward force. The force on the man is
equal in magnitude and opposite in direction to the force on the
woman. Yet the acceleration of the woman is greater than the
acceleration of the man due to the smaller mass of the woman.
Explanation
• The man experienced a backward force. "After all," they might
argue, "the man did not move backward." Such observers are
presuming that forces cause motion. In their minds, a backward
force on the male skater would cause a backward motion.
• Forces cause acceleration, not motion.
Newton’s laws Collisions
• The law of action-reaction (Newton's third law) explains the
nature of the forces between the two interacting objects.
According to the law, the force exerted by object 1 upon object 2
is equal in magnitude and opposite in direction to the force
exerted by object 2 upon object 1.
Conservation of momentum
• 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.
• 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.
• The forces act between the two objects for a given amount of
time.
• 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.
• But the impulse experienced by an object is equal to the change
in momentum of that object (the impulse-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
Law of conservation of energy
• In a collision, the momentum change of object 1 is equal to and
opposite of the momentum change of object 2. That is, the
momentum lost by object 1 is equal to the momentum gained by
object 2.
• In most collisions 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.
• 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.
Go to Momentum questions
• http://www.physicsclassroom.com/class/momentum/Lesson2/Momentum-Conservation-Principle
Isolated system
• A system is a collection of two or more objects. An isolated system
is a system that F is free from the influence of a net external
force that alters the momentum of the system. There are two
criteria for the presence of a net external force; it must be...
• a force that originates from a source other than the two objects of
the system
• a force that is not balanced by other forces
• A system in which the only forces that contribute to the
momentum change of an individual object are the forces acting
between the objects themselves can be considered an isolated
system.
• 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 that 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.
• http://www.physicsclassroom.com/class/momentum/Lesson2/Isolated-Systems
• If a system is not isolated, then the total system momentum is not
conserved.
More problems
• http://www.physicsclassroom.com/class/momentum/Lesson2/Using-Equations-as-a-Recipe-for-Algebraic-Problem
Inelastic collisions
• This lack of conservation means that the forces between colliding
objects may remove or add internal kinetic energy.
• Work done by internal forces may change the forms of energy
within a system.
• For inelastic collisions, such as when colliding objects stick
together, this internal work may transform some internal kinetic
energy into heat transfer.
• Or it may convert stored energy into internal kinetic energy, such
as when exploding bolts separate a satellite from its launch
vehicle.
• Two objects that have equal masses head toward one another at equal
• speeds and then stick together. Their total internal kinetic energy is
initially 1/2 mv^2 +1/2 mv^2 = mv^2 .
• The two objects come to rest after stiking together, conserving
momentum.
• But the internal kinetic energy is zero after the collision.
• A collision in which the objects stick together is sometimes called a
perfectly inelastic collision because it reduces internal kinetic energy
morethan does any other type of inelastic collision. In fact, such a
collision reduces internal kinetic energy to the minimum it can
havewhile still conserving momentum.
Perfectly Inelastic Collision
• A collision in which the objects stick together is sometimes called
“perfectly inelastic.”
Strategy
Elastic Collisions
• A collision which the total momentum and the total kinetic energy
remain constant
Elastic collision equal mass
• Some interesting situations arise when the two colliding objects
have equal mass and the collision is elastic. This situation is nearly
the case with colliding billiard balls, and precisely the case with
some subatomic particle collisions. We can thus get a mental
image of a collision of subatomic particles by thinking about
billiards (or pool).
• First, an elastic collision conserves internal kinetic energy. Again,
let us assume object 2 (m2) is initially at rest. Then, the internal
kinetic energy before and after the collision of two objects that
have equal masses is
• Because the masses are equal, m1 = m2 = m
• Algebraic manipulation (left to the reader) of conservation of
momentum in the x - and y -directions can show that
Rocket Propulsion
• Rockets range in size from fireworks so small that ordinary people use
them to immense Saturn Vs that once propelled massive payloads toward
the Moon.
• The propulsion of all rockets, jet engines, deflating balloons, and even
squids and octopuses is explained by the same physical principle—
Newton’s third law of motion.
• Matter is forcefully ejected from a system, producing an equal and
opposite reaction on what remains.
• Another common example is the recoil of a gun. The gun exerts a force
on a bullet to accelerate it and consequently experiences an equal and
opposite force, causing the gun’s recoil or kick.
Linear Momentum
• Newton actually stated his second law of motion in terms of
momentum: The net external force equals the change in
momentum of a system divided by the time over which it changes.
Using symbols, this law is
• where Fnet is the net external force, Δp is the change in
momentum, and Δt is the change in time.