physics 1501: lecture 17, pg 1 physics 1501: lecture 17 today’s agenda l homework #6: due friday l...

Physics 1501: Lecture 17, Pg 1

Physics 1501: Lecture 17Physics 1501: Lecture 17

TodayToday’’s Agendas Agenda

Homework #6: due Friday

Midterm I: Friday only

TopicsChapter 9

» Momentum

» Introduce Collisions


Chapter 9Chapter 9Linear MomentumLinear Momentum

Newton’s 2nd Law:

FF = maa

dv vp

Units of linear momentum are kg m/s.

pp = mvv(pp is a vector since vv is a

vector).

So px = mvx etc.

Definition: Definition: For a single particle, the momentum pp is defined as:


Momentum ConservationMomentum Conservation

The concept of momentum conservation is one of the most fundamental principles in physics.

This is a component (vector) equation.We can apply it to any direction in which there is no

external force applied. You will see that we often have momentum conservation

even when (mechanical) energy is not conserved.


Elastic vs. Inelastic CollisionsElastic vs. Inelastic Collisions A collision is said to be elastic when energy as well as

momentum is conserved before and after the collision. Kbefore = Kafter

Carts colliding with a spring in between, billiard balls, etc.

vvi

A collision is said to be inelastic when energy is not conserved before and after the collision, but momentum is conserved. Kbefore Kafter

Car crashes, collisions where objects stick together, etc.


Inelastic collision in 1-D: Example 1Inelastic collision in 1-D: Example 1 A block of mass M is initially at rest on a frictionless horizontal surface. A bullet of mass m is fired at the block

with a muzzle velocity (speed) v. The bullet lodges in the block, and the block ends up with a speed V. In terms of m, M, and V :What is the momentum of the bullet with speed v ?What is the initial energy of the system ?What is the final energy of the system ? Is energy conserved ?

v V

before after

x


Example 1...Example 1...

Consider the bullet & block as a system. After the bullet is shot, there are no external forces acting on the system in the x-direction Momentum is conserved in the x direction !Momentum is conserved in the x direction !

Px,before = Px,after

mv = (M+m) V

vV

before after

x


Example 1...Example 1... Now consider the energy of the system before and after:

Before:

After:

So

Energy is NOT conservedEnergy is NOT conserved (friction stopped the bullet)

However momentum was conserved, and this was useful.


Lecture 17,Lecture 17, ACT 1 ACT 1Inelastic Collision in 1-DInelastic Collision in 1-D

ice

(no friction)

Winter in Storrs


Lecture 17,Lecture 17, ACT 1 ACT 1Inelastic Collision in 1-DInelastic Collision in 1-D

vvff = ?

finally

m

v = 0ice

M = 2m

VV00(no friction)

initially

Vf = A) 0 B) Vo/2 C) 2Vo/3 D) 3Vo/2 E) 2Vo


Lecture 17, Lecture 17, ACT 2ACT 2Momentum ConservationMomentum Conservation

Two balls of equal mass are thrown horizontally with the same initial velocity. They hit identical stationary boxes resting on a frictionless horizontal surface. The ball hitting box 1 bounces back, while the ball hitting box 2 gets stuck.

Which box ends up moving fastest ?

(a)(a) Box 1 (b)(b) Box 2 (c)(c) same

1 2


Inelastic collision in 2-DInelastic collision in 2-D

Consider a collision in 2-D (cars crashing at a slippery intersection...no friction).

vv1

vv2

VV

before after

m1

m2

m1 + m2


Inelastic collision in 2-D...Inelastic collision in 2-D... There are no net external forces acting.

Use momentum conservation for both components.

vv1

vv2

VV = (Vx,Vy)

m1

m2

m1 + m2


Inelastic collision in 2-D...Inelastic collision in 2-D...

So we know all about the motion after the collision !

VV = (Vx,Vy)

Vx

Vy


Inelastic collision in 2-D...Inelastic collision in 2-D...

We can see the same thing using vectors:

PP

pp1

pp2

PP

pp1

pp2


Comment on Energy ConservationComment on Energy Conservation

We have seen that the total kinetic energy of a system undergoing an inelastic collision is not conserved.Energy is lost:

» Heat (bomb)

» Bending of metal (crashing cars)

Kinetic energy is not conserved since work is done during the collision !

Momentum along a certain direction is conserved when there are no external forces acting in this direction.In general, easier to satisfy than energy conservation.


Elastic CollisionsElastic Collisions

Elastic means that energy is conserved as well as momentum.

This gives us more constraints.We can solve more complicated problems !!Billiards (2-D collision).The colliding objects

have separate motionsafter the collision as well as before.

Start with a simpler 1-D problem.

Before After


Elastic Collision in 1-DElastic Collision in 1-D

v1,b v2,b

before

x

m1m2

v1,av2,a

afterm1

m2



v1,b v2,b

v1,av2,a

before

after

x

m1 m2

Conserve PX

m1v1,b + m2v2,b = m1v1,a + m2v2,a

Conserve Energy

1/2 m1v21,b + 1/2 m2v2

2,b = 1/2 m1v21,a + 1/2 m2v2

2,a

Suppose we know v1,b and v2,b

We need to solve for v1,a and v2,a

Should be no problem 2 equations & 2 unknowns !

m1v1,b + m2v2,b = m1v1,a + m2v2,a



After some moderately tedious algebra, we can derive the following equations for the final velocities,


Example - Elastic CollisionExample - Elastic Collision

Suppose I have 2 identical bumper cars. One is motionless and the other is approaching it with velocity v1. If they collide elastically, what is the final velocity of each car ?

Note that this means,

m1 = m2 = m

v2B = 0


Example - Elastic CollisionExample - Elastic Collision

Let’s start with the equations for conserving momentum and energy, mv1,B = mv1,A + mv2,A (1)

1/2 mv21,B = 1/2 mv2

1,A + 1/2 mv22,A (2)

Initially, v2 = 0. After the collision, v1 = 0.

To satisfy equation (1), v2A = v1B.

v1A=0 , v2A=0 or both Both: contradicts (2)

If v2A=0 : v1A= v1B from (1) or (2) no collision !


Lecture 17,Lecture 17, ACT 3ACT 3Elastic CollisionsElastic Collisions

I have a line of 3 bumper cars all touching. A fourth car smashes into the others from behind. Is it possible to satisfy both conservation of energy and momentum if two cars are moving after the collision?All masses are identical, elastic collision.

A) Yes B) No

Before

After?


Example of 2-D elastic collisions:Example of 2-D elastic collisions:BilliardsBilliards

If all we know is the initial velocity of the cue ball, we don’t have enough information to solve for the exact paths after the collision. But we can learn some useful things...


BilliardsBilliards

Consider the case where one ball is initially at rest.

ppa

ppb

FF PPa

before afterthe final direction of the

red ball will depend on

where the balls hit.

vvcm


We know momentum is conserved: ppb = ppa + PPa

We also know that energy is conserved:

Comparing these two equations tells us that:

BilliardsBilliards

pb2 = (ppa + PPa )2 = pa

2 + Pa2 + 2 ppa PPa

m2

P

m2

p

m2

p 2a

2a

2 b

ppa PPa = 0

and must therefore be orthogonal!

Or … one momentum must be zero.

ppa

ppb

PPa

Ppp 2a

2a

2 b


BilliardsBilliards

The final directions are separated by 90o.

ppa

ppb

FF PPa

before after

vvcm

Active Figure


Lecture 17 –Lecture 17 – ACT 4ACT 4Pool SharkPool Shark

Can I sink the red ball without scratching ?Ignore spin and friction.

A) Yes B) No C) More info needed


BilliardsBilliards..

More generally, we can sink the red ball without sinking the white ball – fortunately.


BilliardsBilliards

However, we can also scratch. All we know is that the angle between the balls is 90o.

physics 1501: lecture 17, pg 1 physics 1501: lecture 17 today’s agenda l homework #6: due friday l...

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