Chapter 7Momentum and Collisions

MomentumNewton’s Laws give a description of forces

○ There is a force acting or their isn’t○ But what about in between

Constant velocity

Momentum is a quantity defined by the velocity of a mass○ Velocity changes -> Force

Impulse

Momentum describes collisions○ Magnitude gives an indication of how hard it is to

stop○

Momentum From Newton’s laws: force must be present to change an object’s

velocity (speed and/or direction) Wish to consider effects of collisions and corresponding change

in velocity

Method to describe is to use concept of linear momentum

scalar vector

Linear momentum = product of mass velocityLinear momentum = product of mass velocity

Golf ball initially at rest, so some of the KE of club transferred to provide motion of golf ball and its change in velocity

Momentum and Impulse

The linear momentum p of an object of mass m moving with velocity v is defined as

p = mv

Same direction as velocityUnits

○ kg x m / s

m v

p

Momentum

Vector quantity, the direction of the momentum is the same as the velocity’s

Applies to two-dimensional motion as well

yyxx m vpa n dm vp

vmp

Size of momentum: depends upon mass depends upon velocity

Impulse

In order to change the momentum of an object (say, golf ball), 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(F Δt) is defined as the impulseImpulse is a vector quantity, the direction is the same as the

direction of the force

tFporamt

vvm

t

pF n et

ifn et

:)(

Graphical Interpretation of Impulse

Usually force is not constant, but time-dependent

If the force is not constant, use the average force applied

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 time-varying force gives in the interval

( )i

i it

im pu lse F t a rea under F t cu rve

If force is constant: impulse = F t

Example: Impulse Applied to Auto Collisions The most important factor is the collision time

or the time it takes the person to come to a restThis will reduce the chance of dying in a car crash

Ways to increase the timeSeat beltsAir bags

The air bag increases the The air bag increases the time of the collisiontime of the collision and and absorbs some of the energyabsorbs some of the energy from the from the bodybody

ConcepTest

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.

ConcepTest

Suppose a ping-pong ball and a bowling ball are rolling toward you. Both have the same momentum, and you exert the same force to stop each. How do the time intervals to stop them compare?

1. It takes less time to stop the ping-pong ball.2. Both take the same time.3. It takes more time to stop the ping-pong ball.

Note: Because force equals the time rate of change of momentum, the two balls loose momentum at the same rate. If both balls initially had the same momenta, it takes the same amount of time to stop them.

Problem: Teeing Off

A 50-g golf ball at rest is hit by “Big Bertha” club with 500-g mass. After the collision, golf leaves with velocity of 50 m/s.

a) Find impulse imparted to ballb) Assuming club in contact with

ball for 0.5 ms, find average force acting on golf ball

Problem: teeing off

Given:

mass: m=50 g = 0.050 kgvelocity: v=50 m/s

Find:

impulse=?Faverage=?

1. Use impulse-momentum relation:

2. Having found impulse, find the average force from the definition of impulse:

smkg

smkg

mvmvpimpulse if

50.2

050050.0

N

s

smkg

t

pFthustFp

3

3

1000.5

105.0

50.2,

Note: according to Newton’s 3rd law, that is also a reaction force to club hitting the ball:

Momentum and Impulse

Can describe Newton’s laws more precisely with momentum

Law I : If no forces act, Fnet = 0 so a = 0 which implies that velocity is constant p = mv, therefore momentum is constantMomentum constant => conserved quantity

Law II : F = Δp / Δt = m (Δv / Δt) = ma Law III : Rate of change of momentum

generated by action force is exactly opposite to the rate of change of momentum by the reaction force

Conservation of Momentum Definition: an isolated system is the one that

has no external forces acting on it

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

Momentum in an isolated system in Momentum in an isolated system in which a which a collision occurs is conserved (regardless collision occurs is conserved (regardless of the nature of the forces between the of the nature of the forces between the objects)objects)

Conservation of Momentum

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

Conservation of Momentum

Mathematically:

Momentum is conserved for the system of objectsThe system includes all the objects interacting with each otherAssumes only internal forces are acting during the collisionCan be generalized to any number of objects

ffii vmvmvmvm 22112211

Momentum and Impulse

Example:

Conservation of Momentum If the isolated system consists of two

objects undergoing a collision, the total momentum of the system is the same before and after the collision. Same concept as energy

m1v1i + m2v2i = m1v1f + m2v2f

Conservation of Momentum Remember conservation of momentum

applies to the systemThe system includes all the objects

interacting with each otherAssumes only internal forces are acting

during the collision

You must define the isolated systemCan be generalized to any number of

objects

ConcepTest 7.1 Rolling in the Rain

1) speeds up1) speeds up

2) maintains constant speed2) maintains constant speed

3) slows down3) slows down

4) stops immediately4) stops immediately

An open cart rolls along a

frictionless track while it is

raining. As it rolls, what happens

to the speed of the cart as the rain

collects in it? (assume that the

rain falls vertically into the box)

Since the rain falls in vertically, it adds

no momentum to the box, thus the box’s

momentum is conserved. However,

since the mass of the box slowly

increasesincreases with the added rain, its

velocity has to decreasedecrease.

ConcepTest 7.1 Rolling in the Rain

1) speeds up1) speeds up

2) maintains constant speed2) maintains constant speed

3) slows down3) slows down

4) stops immediately4) stops immediately

An open cart rolls along a

frictionless track while it is

raining. As it rolls, what happens

to the speed of the cart as the rain

collects in it? (assume that the

rain falls vertically into the box)

Follow-up:Follow-up: What happens to the cart when it stops raining? What happens to the cart when it stops raining?

Collisions

Momentum is conserved in any collision We can define two types of collisions to

describe how momentum is transferred within the systemElastic

○ Both momentum and kinetic energy are conserved

Inelastic○ Kinetic energy is not conserved, momentum is

conserved

Collisions

Elastic Collisions Momentum is conserved Total Energy is

conserved

2f22

2f11

2i22

2i11

f22f11i22i11

vm2

1vm

2

1vm

2

1vm

2

1

vmvmvmvm

Collisions

Elastic CollisionsInstead of using the energy quadratic

equation, we can simplify the equation to

The relative velocity of the two objects before the collision equals the negative of the relative velocity of the two objects after the collision

)vv(vv f2f1i2i1

Collisions

Elastic CollisionConsider the case: both targets movingSo, m1v1i + m2v2i = m1v1f + m2v2f

½m1v21i + ½ m2v2

2i = ½ m1v21f + ½m2v2

We know m1, m2, v1i, and v2i

Solve for v1f, v2f

Collisions

General Equation

v1f = ————— v1i + ————— v2i

v2f = ————— v1i + ————— v2i

m1 – m2 2m2

2m2 m1 – m2

m1 +m2 m1 + m2

m1 +m2 m1 + m2

Collisions

Consider the case v2i = 0

we get,

v1f = ———— v1i

v2f = ———— v1i

m1 – m2

2m2

m1 + m2

m1 + m2

Collisions (special cases)

Special situations

1. Equal masses (m2 = m1) Exchange velocities

If v2i = 0, v1f = 0 and v2f = v2i

○ body 1 stops and body 2 takes off with speed v1i

Collisions (special cases)

1. Massive target (m2 >> m1) v1f ≈ -v1i and v2f ≈ (2m1 / m2)v1i

body 1 bounces back in opposite direction, body 2 moves slowly

○ Example: golf ball at cannonball

Collisions (special cases)

1. Massive projectile (m1 >> m2) v1f ≈ v1i and v2f ≈ 2v1i

Body 1 continues at same speed, body two at twice the speed!

○ Example: cannonball at golf ball

Collisions Example

Types of Collisions

Momentum is conserved in any collision

what about kinetic energy?

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

Actual collisions Most collisions fall between elastic and perfectly inelastic

collisions

e n e r g ylo s t fi K EK E

Collisions

Perfectly InelasticObjects stick together

○ Not all of the KE is necessarily lost

Most real life collisions fall between elastic and perfectly inelastic collisions

Collisions

Perfectly Inelastic CollisionsSince both masses

have the same velocity, conservation of momentum becomes,

m1v1i + m2v2i = (m1 + m2)vf

Collisions Example of perfectly inelastic collision

Momentum is conserved, kinetic energy is not conserved and converted into heat, sound, and destruction of smaller car

Collisions

Application – Perfectly Inelastic CollisionBallistic Pendulum

○ Initially used to measure speeds of bullets before electronic timing devices were developed.

Sketches for Collision Problems Draw “before” and

“after” sketches Label each object

include the direction of velocity

keep track of subscripts

Sketches for Perfectly Inelastic Collisions The objects stick

together Include all the

velocity directions The “after” collision

combines the masses

Perfectly Inelastic Collisions:

When two objects stick together after the collision, they have undergone a perfectly inelastic collision

Suppose, for example, v2i=0. Conservation of momentum becomes

fii vmmvmvm )( 212211

.20105.2

105

,)2500(0)50)(1000(

:1500,1000ifE.g.,

3

4

21

smkg

smkgv

vkgsmkg

kgmkgm

f

f

fi vmmvm )(0 2111

Perfectly Inelastic Collisions: What amount of KE lost

during collision?

Jsmkg

vmvmKE iibefore

62

222

211

1025.1)50)(1000(2

12

1

2

1

Jsmkg

vmmKE fafter

62

221

1050.0)20)(2500(2

1

)(2

1

JK E l o s t61 07 5.0

lost in heat/”gluing”/sound/…

Collisions Example – Perfectly Inelastic Collision

Two objects, one with mass of 5kg and the other 3kg, approach each other with velocities of 2m/s from opposite directions. After they collide, they stick together. What is the initial and final kinetic energy?

M = 5 kg M = 3 kg

Glancing Collisions

For a general collision of two objects in three-dimensional 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

fyfyiyiy

fxfxixix

vmvmvmvm

andvmvmvmvm

22112211

22112211

Glancing Collisions

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

Two-dimensional Collisions

For a general collision of two objects in three-dimensional 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

fyfyiyiy

fxfxixix

vmvmvmvm

vmvmvmvm

22112211

22112211 and

ffii vmvmvmvm 22112211

Glancing Collisions

Two dimensional collisionsMomentum is still conservedApply vector components to conservation of

momentum equation

ApplicationsBilliard ballsParticle physics

Center of MassIn (a), the diver’s motion is pure translation; in (b) it is translation plus rotation.

There is one point that moves in the same path a particle would take if subjected to the same force as the diver. This point is called the center of mass (CM).

Center of Mass

The general motion of an object can be considered as the sum of the translational motion of the CM, plus rotational, vibrational, or other forms of motion about the CM.

Center of Mass

For two particles, the center of mass lies closer to the one with the most mass

where M is the total mass.

Center of Mass

Example