Chapter 7-Linear Momentum - University of Reginauregina.ca/~barbi/academic/phys109/2009/notes/lecture-17.pdf · Chapter 7 • Momentum and Its Relation to Force • Conservation of

Chapter 7- Linear Momentum

AssignmentAssignment 66Textbook (Giancoli, 6 th edition), Chapter 6:

Due on Thursday, November 12

1. On page 163 of Giancoli, problem 38.

2. On page 165 of Giancoli, problem 69.

3. On page 188 of Giancoli, problem 12.3. On page 188 of Giancoli, problem 12.

4. A gymnast of mass 52 kg is jumping on a trampoline. She jumps so that her feet reach a maximum height of 2:5 m above the trampoline and, when she lands, her feet stretch the trampoline down 75 cm. How far does the trampoline stretch when she stands on it at rest? [Hint: Assume the trampoline obeys Hook's law when it is stretched.]

Chapter 7

• Momentum and Its Relation to Force

• Conservation of Momentum

• Collisions and Impulse• Collisions and Impulse

• Conservation of Energy and Momentum in Collisions

• Elastic Collisions in One Dimension

• Inelastic Collisions

• Collisions in Two or Three Dimensions

Recalling Recalling LastLast LectureLectureRecalling Recalling LastLast LectureLecture

When the work done by a force does NOT depend on the path taken, this FORCE is said to be CONSERVATIVE.

When the work done by a force DOES depend on the path taken, this FORCE is said to be NONCONSERVATIVE.

Nonconservative forces do NOT have a potential energy associated to them.

Conservative and Conservative and NonconservativeNonconservative ForcesForces

Thus, potential energy can be defined only for cons ervative forces.

Eq. 6-14 tells us that the work done by nonconserva tive forces is equal to the total change in kinetic and potential energies.

(6-14)

“ If only conservative forces are acting, the TOTAL MECHANICAL ENERGY of a system neither increases nor decreases in any proce ss. It stays constant �� it is CONSERVED”

POWER:

Power is the rate at which work is done, or the rate at which energy is

Mechanical Energy and its ConservationMechanical Energy and its Conservation

Power is the rate at which work is done, or the rate at which energy is transformed .

In the SI system the unit of power is: 1 hp = 746 W

In general, if an object moves with average velocity , the power can be written as

(6-18)

(6-19)

Kinetic Energy, and the Work Energy PrincipleKinetic Energy, and the Work Energy Principle

Problem 6-63 (textbook): A driver notices that her 1150-kg car slows down from 85 Km/h to 65 Km/h in about 6.0 s on the level when it is in neutral. Approximately what power (watts and hp) is needed to keep the car traveling at a constant 75 km/h?

Kinetic Energy, and the Work Energy PrincipleKinetic Energy, and the Work Energy Principle

Problem 6-63

The energy transfer from the engine must replace the lost kinetic energy. From the two speeds, calculate the average rate of loss in kinetic energy while in neutral.

( ) ( ) ( )

1 2

2 22 2 51 1 12 12 2 2

1m s 1m s85 km h 23.61m s 65 km h 18.06 m s

3.6 km h 3.6 km h

1150 kg 18.06 m s 23.61m s 1.330 10 J

v v

KE mv mv

= = = =

∆ = − = − = − ×

Note now that 75 Km/h is the average between the car’s initial and final speeds:

the is needed from the engine.

( ) ( ) ( )

( )2 12 2 2

54 4

1150 kg 18.06 m s 23.61m s 1.330 10 J

1.330 10 J 1 hp2.216 10 W , or 2.216 10 W 29.71 hp

6.0 s 746 W

KE mv mv

WP

t

∆ = − = − = − ×

×= = = × × =

⇒⇒⇒⇒ We can use to conclude

4 12.2 10 W or 3.0 10 hp× ×

TodayTodayTodayToday

The momentum of an object tells how hard (or easy) is to change its state of motion.

Example: It is easier to stop a car when it is moving at 10 km/h than when it is moving at 100 Km/h.

� But, note that it will also depend on the mass of the car: a heavy truck moving at

Momentum and Its Relation to ForceMomentum and Its Relation to Force

� But, note that it will also depend on the mass of the car: a heavy truck moving at 10 Km/h is more difficult to be brought to rest than a small Mercedes SMART (which is much lighter than a truck) moving at the same speed.

Momentum is a vector symbolized by the symbol p, and is defined as

(6-20)

Recalling that acceleration is the rate at which velocity changes, we can then write:

But,

Defining: and , we can write


The right side of the equation 6.21 is the net force acting on the object. Then:

(6-21)

(6-22)

Eq. 6-22 is another way of expressing Newton’s second law. However, it is a more general definition because it introduces the situation where the mass may change.

Example: A rocket burns fuel when lifting off or maneuvering in space:

� As it burns fuel, the rocket becomes lighter


(6-22)

� So, its total mass changes and so does its momentum (even if you manage to keep its velocity constant)

Collisions are common event in everyday life.

During a collision, objects are deformed due to the large forces involved.

These forces are generally very strong, much stronger than other interactions between the colliding objects and their surrounding environment, and act for a very short period of time ∆t.

We can use eq. 6-22 and define the impulse on an object as:

Collision and ImpulseCollision and Impulse

In general, the forces involved in the process are not constantduring ∆t, but we can approximate the resulting force by the average force acting during this interval of time:

(6-23)

(6-24)

Let’s consider a collision between two billiard balls as shown in the figure.

Conservation of MomentumConservation of Momentum

Assume that the two balls form a system isolatedfrom the external world:

� in other words � there is NO net force acting on the billiard balls other than the interaction between them.

The momentum of each ball before and after the collision is:

Object A: before � , after �

Object B: before � , after �

Recalling previous slide: The momentum of each ball before and after the collision is:


Object A: ,

Object B: ,

Now, ball A exerts a force on ball B.

According to the general form of Newton’s second law:

But,

On the other hand, according to Newton’s third law, B exerts a force

We have then:


Combining the two equations, we find:

(6-25)

Equation 6-25 tells that the total momentum of the system (the sum of the momentum of the two balls) before the collision is equal to the total momentum of the system after the collision IF the external net force acting on the system is zero �� isolated system . This is known as Conservation of Total Momentum .

The above equation can be extended to include any number of objects such that the only forces are the interaction between the objects in the system.


Momentum conservation works for a rocket as long as we consider the rocket and its fuel to be one system, and account for the mass loss of the rocket.

Here you can consider a frame at rest relative to the rocket before it lifts off

� Its initial momentum is then zero. As it takes off, the fuel burns expelling gas in one direction

� this gives momentum to the gas

� in order to obey momentum conservation, the rocket has to move in opposite direction such that the total momentum (rocket + gas) remains zero.

Conservation of Energy and Momentum in CollisionsConservation of Energy and Momentum in Collisions

In general, we can identify two different types of collisions:

1. Elastic collision2. Inelastic collision

In elastic collisions the total kinetic energy of a system is conserved .� No energy dissipates in form of heat or other form of energy.

An example is the collision between the two billiard balls discussed in the previous slides:balls discussed in the previous slides:

In inelastic collisions , there is NO conservation ofkinetic energy .

� Some of the total initial kinetic energy is transformed into some other form of energy.

(6-26)


Example of inelastic collision:

With inelastic collisions, some of the initial kinetic energy is lost to thermal or potential energy. It may also be gained during explosions, as there is the addition of chemical or nuclear energy.

A completely inelastic collision is one where the objects stick together afterwards, so there is only one final velocity.final velocity.


Note:

Regardless whether we have inelastic or elastic collisions, the total momentum is always conserved if the system is isolated.

For instance, the collision between two trains as depicted below is inelastic. Part of the total initial kinetic energy might have been transformed into thermal or some other form of energy.

However, the total momentum of the closed (isolated) system (two trains) should be However, the total momentum of the closed (isolated) system (two trains) should be conserved.

(The textbookhas a goodexample of thisproblem).


Note:

Also…

The total energy (the sum of all energies) in a clo sed (isolated system) is ALWAYS conserved.is ALWAYS conserved.


Elastic collision in one dimension:

Here we have two objects colliding elastically. We know the masses and the initial speeds.

Since both total momentum and kinetic energy are conserved , we can write two equations . This allows us to solve for the two unknown final speeds .

From conservation of momentum:

We can rearrange this equation and write:



Since it is an elastic collision, the total kinetic energy is also conserved:

using that , then

We then have two equations from total momentum and kinetic energy conservation:

Replacing the last equation into the first one:



Eq. 6-27 tell us that in ONE DIMENSION elastic head-oncollision, the relative velocity between the objects have the same magnitude but in opposite direction before and after the collision.

(6-27)


Collision in two or more dimensions

That is just a generalization of what we have discussed in the previous slides.

We have to make use of the concept of vector components.

In this particular case, the reference system is taken such B is initially (before the collision) at rest and A moves in the x direction before the collision.

In the figure, considering the system isolated, the total momentum has to be conserved in both x and y directions.


Collision in two or more dimensions

We then have:

(i)

(ii)

It follows then, using the above expressions into (i) and (ii), that:

⇒ This gives a system of two

equations and three variables.

If you can measure any of these

variables, the other two can be

calculated from system of equations.

Linear MomentumLinear Momentum

Problem 7-1 (textbook) A constant friction force of 25 N acts on a 65-kg skier for 20 s. What is the skier’s change in velocity?


Problem 7-1 :

From Newton’s second law,

For a constant mass object,

t∆ = ∆p Frr

m∆ = ∆p vr r

Equate the two expressions for

If the skier moves to the right, then the speed will decrease, because the friction force is to the left.

The skier loses 7.7 m/s of speed.

∆ pr

t

t mm

∆∆ = ∆ → ∆ =

FF v v

rr r r

( ) ( )25 N 20 s7 .7 m s

65 kg

F tv

m

∆∆ = − = − = −


Problem 7-4 (textbook) A child in a boat throws a 6.40-kg package out horizontally with a speed of 10.0 m/s. Calculate the velocity of the boat immediately after, assuming it was initially at rest. The mass of the child is 26.0 kg, and that of the boat is 45.0 kg. Ignore water resistance.


Problem 7-4 :

The throwing of the package is a momentum-conserving action, if the water resistance is ignored.

Let “A” represent the boat and child together, and let “B” represent the package.

Choose the direction that the package is thrown as the positive direction. Apply conservation of momentum, with the initial velocity of both objects being 0.

The boat and child move in the opposite direction as the thrown package.

( )( )( )( )

i n i t i a l f in a l

6 .4 0 k g 1 0 .0 m s0 .9 0 1 m s

2 6 .0 k g 4 5 .0 k g

A B A A B B

B BA

A

p p m m v m v m v

m vv

m

′ ′= → + = + →

′′ = − = − = −

+


Problem 7-34 (textbook) An internal explosion breaks an object, initially at rest, into two pieces, one of which has 1.5 times the mass of the other. If 7500 J were released in the explosion, how much kinetic energy did each piece acquire? (Assume that all available released energy is carried away by the two particles)..


Problem 7-34

Use conservation of momentum in one dimension, since the particles will separate and travel in opposite directions. Call the direction of the heavier particle’s motion the positive direction. Let A represent the heavier particle, and B represent the lighter particle. We have

A B1.5m m=A B 0v v= =

B B 2 0 m v

p p m v m v v v′

′ ′ ′ ′= → = + → = − = −

The negative sign indicates direction. Since there was no mechanical energy before the explosion, the kinetic energy of the particles after the explosion must equal the energy added.

Thus:

2initial final A A B B A B3

A

0 p p m v m v v vm

′ ′ ′ ′= → = + → = − = −

( ) ( ) ( )( )

22 2 2 25 51 1 1 2 1 1added 2 2 2 3 2 3 2 3

3 3added added5 5

1.5

7500 J 4500 J 7500 J 4500 J 3000 J

A B A A B B B B B B B B B

B A B

E KE KE m v m v m v m v m v KE

KE E KE E KE

′ ′ ′ ′ ′ ′ ′ ′= + = + = + = =

′ ′ ′= = = = − = − =

3 33.0 10 J 4.5 10 JA BK E KE′ ′= × = ×

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Chapter 7-Linear Momentum - University of Reginauregina.ca/~barbi/academic/phys109/2009/notes/lecture-17.pdf · Chapter 7 • Momentum and Its Relation to Force • Conservation of