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 The linear momentum of an object of mass m moving with a velocity v is
defined as the product of the mass and the velocity
 p = m v
 SI Units are kg m / s
 Vector quantity, the direction of the momentum is the same as the
velocity’s

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 Applies to twodimensional motion

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 In order to change the momentum of an object, a force must be applied
 The time rate of change of momentum of an object is equal to the net
force acting on it
 Gives an alternative statement of Newton’s second law

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 When a single, constant force acts on the object
 FΔt is defined as the impulse
 Vector quantity, the direction is the same as the direction of the
force

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 The theorem states that the impulse acting on the object is equal to the
change in momentum of the object
 If the force is not constant, use the average force applied

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 The average force can be thought of as the constant force that would
give the same impulse to the object in the time interval as the actual
timevarying force gives in the interval

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 The impulse imparted by a force during the time interval Δt is
equal to the area under the forcetime graph from the beginning to the
end of the time interval
 Or, to the average force multiplied by the time interval

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 The most important factor is the collision time or the time it takes the
person to come to a rest
 This will reduce the chance of dying in a car crash
 Ways to increase the time

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 The air bag increases the time of the collision
 It will also absorb some of the energy from the body
 It will spread out the area of contact
 decreases the pressure
 helps prevent penetration wounds

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 Momentum in an isolated system in which a collision occurs is conserved
 A collision may be the result of physical contact between two objects
 “Contact” may also arise from the electrostatic interactions of the
electrons in the surface atoms of the bodies
 An isolated system will have not external forces

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 The principle of conservation of momentum states when no external forces
act on a system consisting of two objects that collide with each other,
the total momentum of the system before the collision is equal to the
total momentum of the system after the collision

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 Mathematically:
 Momentum is conserved for the system of objects
 The system includes all the objects interacting with each other
 Assumes only internal forces are acting during the collision
 Can be generalized to any number of objects

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 The total momentum of an isolated system of objects is conserved
regardless of the nature of the forces between the objects.
 Newton’s third law is a special application of the conservation of
momentum.

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 Momentum is conserved in any collision
 Inelastic collisions
 Kinetic energy is not conserved
 Some of the kinetic energy is converted into other types of energy
such as heat, sound, work to permanently deform an object
 Perfectly inelastic collisions occur when the objects stick together
 Not all of the KE is necessarily lost

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 Elastic collision
 both momentum and kinetic energy are conserved
 Actual collisions
 Most collisions fall between elastic and perfectly inelastic collisions

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 When two objects stick together after the collision, they have undergone
a perfectly inelastic collision
 Conservation of momentum becomes

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 Momentum is a vector quantity
 Direction is important
 Be sure to have the correct signs

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 Both momentum and kinetic energy are conserved
 Typically have two unknowns
 Solve the equations simultaneously

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 A simpler equation can be used in place of the KE equation

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 Set up a coordinate axis and define the velocities with respect to this
axis
 It is convenient to make your axis coincide with one of the initial
velocities
 In your sketch, draw all the velocity vectors with labels including all
the given information

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 Draw “before” and “after” sketches
 Label each object
 include the direction of velocity
 keep track of subscripts

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 The objects stick together
 Include all the velocity directions
 The “after” collision combines the masses

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 Write the expressions for the momentum of each object before and after
the collision
 Remember to include the appropriate signs
 Write an expression for the total momentum before and after the
collision
 Remember the momentum of the system is what is conserved

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 If the collision is inelastic, solve the momentum equation for the
unknown
 Remember, KE is not conserved
 If the collision is elastic, you can use the KE equation (or the
simplified one) to solve for two unknowns

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 For a general collision of two objects in threedimensional space, the
conservation of momentum principle implies that the total momentum of
the system in each direction is conserved
 Use subscripts for identifying the object, initial and final, and
components

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 The “after” velocities have x and y components
 Momentum is conserved in the x direction and in the y direction
 Apply separately to each direction

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 Vector diagram showing
 collision.

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 Since the masses are equal we can eliminate them from our problem.
 cos 40 = V’r/Vb sin 40 = V’b/Vb
 V’r = cos 40 (.80 m/s) V’b = sin 40 (.80)
 V’r = .77 (.80 m/s) V’b =.64(.80 m/s)
 V’r = .62 m/s V’b = .51 m/s

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 Set up coordinate axes and define your velocities with respect to these
axes
 It is convenient to choose the x axis to coincide with one of the
initial velocities
 In your sketch, draw and label all the velocities and include all the
given information

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 Write expressions for the x and y components of the momentum of each
object before and after the collision
 Write expressions for the total momentum before and after the collision
in the xdirection
 Repeat for the ydirection

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 Solve for the unknown quantities
 If the collision is inelastic, additional information is probably
required
 If the collision is perfectly inelastic, the final velocities of the
two objects is the same
 If the collision is elastic, use the KE equations to help solve for the
unknowns

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 The operation of a rocket depends on the law of conservation of momentum
as applied to a system, where the system is the rocket plus its ejected
fuel
 This is different than propulsion on the earth where two objects exert
forces on each other
 road on car
 train on track

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 The rocket is accelerated as a result of the thrust of the exhaust gases
 This represents the inverse of an inelastic collision
 Momentum is conserved
 Kinetic Energy is increased (at the expense of the stored energy of the
rocket fuel)

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 The initial mass of the rocket is M + Δm
 M is the mass of the rocket
 m is the mass of the fuel
 The initial velocity of the rocket is v

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 The rocket’s mass is M
 The mass of the fuel, Δm, has been ejected
 The rocket’s speed has increased to v + Δv

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 The basic equation for rocket propulsion is:
 M_{i} is the initial mass of the rocket plus fuel
 M_{f} is the final mass of the rocket plus any remaining fuel
 The speed of the rocket is proportional to the exhaust speed

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 The thrust is the force exerted on the rocket by the ejected exhaust
gases
 The instantaneous thrust is given by
 The thrust increases as the exhaust speed increases and as the burn
rate (ΔM/Δt) increases

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