Momentum, impulse, and collisions

Momentum, impulse, and collisionsChapter 8

Sections 1-5

Physics Chapter 8 2

Momentum The linear momentum of an object is

defined as

vmp

Momentum has a magnitude of mv and the same direction as v

Physics Chapter 8 3

Impulse

When a particle is acted on by a constant force, the impulse of the force is defined as

tFJ

Physics Chapter 8 4

Impulse-momentum theorem

For a constant force

atvv 0

matmvmv 0

pJ

Ftmvmv 0

0mvmvFt vmtF

Physics Chapter 8 5

Example

A 45-g golf ball initially at rest is given a speed of 25.0 m/s when a club strikes. If the club and ball are in contact for 2.00 ms, what average force acts on the ball?

562 N

Physics Chapter 8 6

On your own

A 0.160-kg hockey puck is moving on an icy, frictionless, horizontal surface. At t = 0 the puck is moving to the right at 3.00 m/s. Calculate the velocity of the puck after a force of 12.0 N directed to the left has been applied for 0.050 s.

0.75 m/s to the left

Physics Chapter 8 7

Conservation of momentum

If the vector sum of the external forces on a system is zero, the total momentum of the system is conserved.

afterbefore pp

21 pp

Physics Chapter 8 8

Example

A runaway 14,000 kg railroad car is rolling horizontally at 4 m/s toward a switchyard. As it passes by a grain elevator, 2000 kg of grain are suddenly dropped into the car. How long does it take the car to cover the 500-m distance from the elevator to the switchyard? Neglect friction and air drag.

143 s

Physics Chapter 8 9

On your own During repair of the Hubble Space

Telescope, an astronaut replaces a damaged solar panel. Pushing the detached panel away into space, she is propelled in the opposite direction. The astronaut’s mass is 60 kg and the panel’s mass is 80 kg. The astronaut is at rest relative to the spaceship when she shoves away the panel, and she shoves it at 0.3 m/s relative to the spaceship. What is her subsequent velocity relative to the spaceship?

-0.4 m/s

Physics Chapter 8 10

Components of momentum

We can separate momentum into components, just like we can with force or velocity

xx mvp yy mvp


Example

A B

30°

90°


Example

A 3.0-kg ball, A, is moving at a velocity of 5.0 m/s. It collides with a stationary ball, B, of mass 2.0 kg. After the collision, ball A moves off in a direction of 30° to the left of its original direction. Ball B moves off in a direction of 90° to the right of ball A’s final direction. Find the speeds of the balls after the collision.

vA=4.33 m/s, vB= 3.75m/s


On your own

A B

30°

45°


On your own A hockey puck B rests on a smooth ice

surface and is struck by a second puck, A, which was originally traveling at 40.0 m/s and which is deflected 30.0° from its original direction. Puck B acquires a velocity at a 45.0° angle to the original direction of A. The pucks have the same mass.

Compute the speed of each puck after the collision.

vA=29.3 m/s vB=20.7 m/s


Elastic collisions

If all the forces acting during a collision are conservative, then no mechanical energy is lost or gained in the collision.

When the total kinetic energy after the collision is the same as it was before the collision, then the collision was elastic.

Collisions between billiard balls, marbles, or other similar objects are nearly elastic.


Inelastic collisions

Collisions between automobiles are one example of inelastic collisions.

The kinetic energy is not conserved, because some energy goes into crumpling the cars.

Collisions in which the two objects stick together afterwards are inelastic.


Example

In a feat of public marksmanship, you fire a bullet of mass mb into a hanging wood block of mass mw. The block, with the bullet embedded, swings upward. Noting the height, h, reached at the top of the swing, you inform the crowd of the bullet’s speed. How fast was the bullet traveling before it hit the block?

((m1+m2)/m1)*sqrt(2gh)


On your own

In Dallas, the morning after a winter ice storm, a 1400-kg automobile going west at 35.0 km/h collides with a 2800-kg truck going south at 50.0 km/h. If they become coupled on collision, what are the magnitude and direction of their velocity after colliding?

35.3 km/h @ 19.3 degrees west of south


Elastic collisions

For elastic collisions, both p and K are conserved.

2211 BBAABBAA vmvmvmvm 1

21

21

22

22 2

1

2

1

2

1

2

1BBAABBAA vmvmvmvm 2


Elastic collisions

2112 AAABBB vvmvvm 1

2112 AAAABBBB vmvmvmvm 1


Elastic collisions

22

21

21

22 AAABBB vvmvvm 2

22

21

21

22 2

1

2

1

2

1

2

1AAAABBBB vmvmvmvm 2

21211212 AAAAABBBBB vvvvmvvvvm 2


Elastic collisions

21

2121

12

1212

AAA

AAAAA

BBB

BBBBB

vvm

vvvvm

vvm

vvvvm

2÷1

2112 AABB vvvv


Be careful!

The last equation is only true for elastic collisions.

It is a special case, not the general case. For all collisions, momentum is conserved,

but kinetic energy is only conserved for elastic collisions.

Do not assume that a collision is elastic unless you are told that it is.


On your own A neutron of mass mn and speed vn1 collides

elastically with a carbon nucleus of mass mc initially at rest. What are the final velocities of both particles?

Be careful! - The velocity equation developed for elastic collisions isn’t enough to solve the problem. It has 2 unknowns, so you need another equation. Use the conservation of momentum equation.

Vn2=-((mc-mn)/(mn+mc))*vn1

Vc2=((2mn)/(mn+mc))*vn1


Center of mass The center of mass of a system of

particles is a weighted average of the position of the particles

If we have several particles with masses m1, m2, etc. and coordinates (x1 , y1), (x2 , y2), etc.

The center of mass of the system is defined as the point with the following coordinates.


Center of mass

ii

iii

m

xm

mmm

xmxmxmX

...

...

321

332211

ii

iii

m

ym

mmm

ymymymY

...

...

321

332211


Example

Find the center of mass of the Earth-Sun system.

The distance between their centers is 1.49 x 1011 m.

The mass of the Sun is 1.99 x 1030 kg. The mass of the Earth is 5.98 x 1024 kg.


Example, continued

The radius of the sun is 6.95 x 108 m

ii

iii

m

xm

mmm

xmxmxmX

...

...

321

332211

kg 1098.5kg 1099.1

m 1049.1kg 1098.5m 0kg 1099.12430

112430

X

m 1048.4 5X


Velocity of the center of mass

Where M = the total mass

M

vmV i

ii

ii

iixi

x m

vmV

ii

iiyi

y m

vmV


Momentum of the center of mass

Total momentum equals the total mass times the velocity of the center of mass.

i

iivmVMP


Acceleration of the Center of Mass

M

amA i

ii


Newton’s second law

All internal forces between the particles cancel out

AMFext


See page 196

The center of mass of the wrench moves in a straight line.

The center of mass of the shell fragments follows the parabolic trajectory of the intact shell.

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Momentum, impulse, and collisions