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9. Systems of Particles. Center of Mass Momentum Kinetic Energy of a System Collisions Totally Inelastic Collisions Elastic Collisions. As the skier flies through the air, most parts of his body follow complex trajectories. But one special point follows a parabola.

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9 systems of particles

9. Systems of Particles

Center of Mass


Kinetic Energy of a System


Totally Inelastic Collisions

Elastic Collisions

As the skier flies through the air,

most parts of his body follow complex trajectories.

But one special point follows a parabola.

What’s that point, and why is it special?

Ans. His center of mass (CM)

Rigid body: Relative particle positions fixed.

9 1 center of mass
9.1. Center of Mass

N particles:

= total mass

= Center of mass

= mass-weighted average position


3rd law 

Cartesian coordinates:

Extension: “particle” i may stand for an extended object with cm at ri.

Example 9 1 weightlifting
Example 9.1. Weightlifting

Find the CM of the barbell consisting of 50-kg & 80-kg weights at opposite ends of a 1.5 m long bar of negligible weight.

CM is closer to the heavier mass.

Example 9 2 space station
Example 9.2. Space Station

A space station consists of 3 modules arranged in an equilateral triangle, connected by struts of length L & negligible mass.

2 modules have mass m, the other 2m.

Find the CM.

Coord origin at m2 = 2m & y points downward.

2: 2m





obtainable by symmetry

1: m



Continuous distributions of matter
Continuous Distributions of Matter

Discrete collection:

Continuous distribution:

Let be the density of the matter.

Example 9 3 aircraft wing
Example 9.3. Aircraft Wing

A supersonic aircraft wing is an isosceles triangle of length L, width w, and negligible thickness.

It has mass M, distributed uniformly.

Where’s its CM?

Density of wing = .

Coord origin at leftmost tip of wing.

By symmetry,


















A high jumper clears the bar,

but his CM doesn’t.

Got it 9 1
Got it? 9.1.

A thick wire is bent into a semicircle.

Which of the points is the CM?

Example 9 4 circus train
Example 9.4. Circus Train

Jumbo, a 4.8-t elephant, is standing near one end of a 15-t railcar,

which is at rest on a frictionless horizontal track.

Jumbo walks 19 m toward the other end of the car.

How far does the car move?

1 t = 1 tonne

= 1000 kg

Final distance of Jumbo from xc:

Jumbo walks, but the center of mass doesn’t move (Fext = 0 ).

9 2 momentum
9.2. Momentum

Total momentum:

M constant 

Conservation of momentum
Conservation of Momentum

Conservation of Momentum:

Total momentum of a system is a constant if there is no net external force.

Got it 9 2
GOT IT! 9.2.

A 500-g fireworks rocket is moving with velocity v = 60 j m/s at the instant it explodes.

If you were to add the momentum vectors of all its fragments just after the explosion,

what would you get?

K.E. is not conserved.

Emech = K.E. + P.E. grav is not conserved.

Etot = Emech + Uchem is conserved.

Conceptual example 9 1 kayaking
Conceptual Example 9.1. Kayaking

Jess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water.

Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water.

What’s the kayak’s speed after Nick catches it?

Why can you answer without doing any calculations ?

Initially, total p = 0.

frictionless water  p conserved

After Nick catches it , total p = 0.

Kayak speed = 0

Simple application of the conservation law.

Making the connection
Making the Connection

Jess (mass 53 kg) & Nick (mass 72 kg) sit in a 26-kg kayak at rest on frictionless water.

Jess toss a 17-kg pack, giving it a horizontal speed of 3.1 m/s relative to the water.

What’s the kayak’s speed while the pack is in the air ?


While pack is in air:

Note: Emech not conserved

Example 9 5 radioactive decay
Example 9.5. Radioactive Decay

A lithium-5 ( 5Li ) nucleus is moving at 1.6 Mm/s when it decays into

a proton ( 1H, or p ) & an alpha particle ( 4He, or  ). [ Superscripts denote mass in AMU ]

 is detected moving at 1.4 Mm/s at 33 to the original velocity of 5Li.

What are the magnitude & direction of p’s velocity?

Before decay:

After decay:

Example 9 6 fighting a fire
Example 9.6. Fighting a Fire

A firefighter directs a stream of water to break the window of a burning building.

The hose delivers water at a rate of 45 kg/s, hitting the window horizontally at 32 m/s.

After hitting the window, the water drops horizontally.

What horizontal force does the water exert on the window?

Momentum transfer to a plane  stream:

= Rate of momentum transfer to window

= force exerted by water on window

Got it 9 3
GOT IT? 9.3.

  • Two skaters toss a basketball back & forth on frictionless ice.

  • Which of the following does not change:

  • momentum of individual skater.

  • momentum of basketball.

  • momentum of the system consisting of one skater & the basketball.

  • momentum of the system consisting of both skaters & the basketball.

Application: Rockets


9 4 collisions
9.4. Collisions

  • Examples of collision:

  • Balls on pool table.

  • tennis rackets against balls.

  • bat against baseball.

  • asteroid against planet.

  • particles in accelerators.

  • galaxies

  • spacecraft against planet ( gravity slingshot )

  • Characteristics of collision:

  • Duration: brief.

  • Effect: intense

  • (all other external forces negligible )

Momentum in collisions
Momentum in Collisions

External forces negligible  Total momentum conserved

For an individual particle

t = collision time


More accurately,

Same size


Crash test

Energy in collisions
Energy in Collisions

Elastic collision: K conserved.

Inelastic collision: K not conserved.

Bouncing ball: inelastic collision between ball & ground.

Got it 9 4
GOT IT? 9.4.

  • Which of the following qualifies as a collision?

  • Of the collisions, which are nearly elastic & which inelastic?

  • a basketball rebounds off the backboard.

  • two magnets approach, their north poles facing; they repel & reverse direction without touching.

  • a basket ball flies through the air on a parabolic trajectory.

  • a truck crushed a parked car & the two slide off together.

  • a snowball splats against a tree, leaving a lump of snow adhering to the bark.





9 5 totally inelastic collisions
9.5. Totally Inelastic Collisions

Totally inelastic collision: colliding objects stick together

 maximum energy loss consistent with momentum conservation.

Example 9 7 hockey
Example 9.7. Hockey

A Styrofoam chest at rest on frictionless ice is loaded with sand to give it a mass of 6.4 kg.

A 160-g puck strikes & gets embedded in the chest, which moves off at 1.2 m/s.

What is the puck’s speed?

Example 9 8 fusion
Example 9.8. Fusion

Consider a fusion reaction of 2 deuterium nuclei 2H + 2H  4He .

Initially, one of the 2H is moving at 3.5 Mm/s, the other at 1.8 Mm/s at a 64 angle to the 1st.

Find the velocity of the Helium nucleus.

Example 9 9 ballistic pendulum
Example 9.9. Ballistic Pendulum

The ballistic pendulum measures the speeds of fast-moving objects.

A bullet of mass m strikes a block of mass M and embeds itself in the latter.

The block swings upward to a vertical distance of h.

Find the bullet’s speed.


(heat is generated when bullet strikes block)

9 6 elastic collisions
9.6. Elastic Collisions

Momentum conservation:

Energy conservation:

Implicit assumption: particles have no interaction when they are in the initial or final states. ( Ei = Ki )

2-D case:

number of unknowns = 2  2 = 4 ( final state: v1fx , v1fy , v2fx , v2fy )

number of equations = 2 +1 = 3

 1 more conditions needed.

3-D case:

number of unknowns = 3  2 = 6 ( final state: v1fx , v1fy , v1fz , v2fx , v2fy , v2fz )

number of equations = 3 +1 = 4

 2 more conditions needed.

Elastic collisions in 1 d
Elastic Collisions in 1-D

1-D collision

1-D case:

number of unknowns = 1  2 = 2 ( v1f , v2f )

number of equations = 1 +1 = 2

 unique solution.

This is a 2-D collision

(a) m1 << m2 :

(b) m1=m2 :

(c) m1 >> m2 :

Example 9 10 nuclear engineering
Example 9.10. Nuclear Engineering

Moderator slows neutrons to induce fission.

A common moderator is heavy water ( D2O ).

Find the fraction of a neutron’s kinetic energy that’s transferred to an initially stationary D in a head-on elastic collision.

Got it 9 5
GOT IT? 9.5.

One ball is at rest on a level floor.

Another ball collides elastically with it & they move off in the same direction separately.

What can you conclude about the masses of the balls?

1st one is lighter.

Elastic collision in 2 d
Elastic Collision in 2-D

Impact parameter b :

additional info necessary to fix the collision outcome.

Example 9 11 croquet
Example 9.11. Croquet

A croquet ball strikes a stationary one of equal mass.

The collision is elastic & the incident ball goes off 30 to its original direction.

In what direction does the other ball move?

p cons:

E cons: