Chapter 6Chapter 6

Momentum and Collisions

MomentumMomentumDefinition:

vmp

Important because it is CONSERVEDproof:

ptF

tp

tv

mF

Since F12=-F21, 021 pp

for isolated particles never changes! ip

Vector Vector quantityquantity

Both px and py are conserved

yy

xxmvpmvp

Example 6.1Example 6.1

An astronaut of mass 80 kg pushes away from a space station by throwing a 0.75-kg wrench which moves with a velocity of 24 m/s relative to the original frame of the astronaut. What is the astronaut’s recoil speed?

Example 6.1Example 6.1

An astronaut of mass 80 kg pushes away from a space station by throwing a 0.75-kg wrench which moves with a velocity of 24 m/s relative to the original frame of the astronaut. What is the astronaut’s recoil speed? 0.225 m/s

Center of mass does not accelerateCenter of mass does not accelerate

Constant

Constant0

0

21

2211

2211

2211

2211

mmrmrmrmrmrmrm

vmvm

If total momentum is zero,

Do the back-to-back demoDo the back-to-back demo

Example 6.2Example 6.2Ted and his ice-boat (combined mass = 240 kg) rest on the frictionless surface of a frozen lake. A heavy rope (mass of 80 kg and length of 100 m) is laid out in a line along the top of the lake. Initially, Ted and the rope are at rest. At time t=0, Ted turns on a wench which winds 0.5 m of rope onto the boat every second.

a) What is Ted’s velocity just after the wench turns on?

b) What is the velocity of the rope at the same time?

c) What is the Ted’s speed just as the rope finishes?

d) What was Ted’s acceleration during this time?

e) How far did Ted move?

f) How far did the center-of-mass of the rope move?

g) How far did the center-of-mass of Ted+boat+rope move?

0.125 m/s

-0.375 m/s

0

-6.25x10-4 m/s2

12.5 m

-37.5 m

0

Example 6.3Example 6.3

A 1967 Corvette of mass 1450 kg moving with a velocity of 100 mph (= 44.7 m/s) slides on a slick street and collides with a Hummer of mass 3250 kg which is parked on the side of the street. The two vehicles interlock and slide off together. What is the speed of the two vehicles immediately after they join?

13.8 m/s =30.9 mph

ImpulseImpulse

Useful for sudden changes when not interested in force but only effects of force

ptF Impulse

Bunjee Jumper DemoBunjee Jumper Demo

Graphical Representation of ImpulseGraphical Representation of Impulse

For a complicated force, impulse is represented graphically by the area under the F vs.t curve.

ptF Impulse

F

t

Total Impulse

t

F

Example Example 6.46.4 A pitcher throws a 0.145-kg

baseball so that it crosses home plate horizontally with a speed of 40 m/s. It is hit straight back at the pitcher with a final speed of 50 m/s.

a) What is the impulse delivered to the ball?

b) Find the average force exerted by the bat on the ball if the two are in contact for 2.0 x 10–3 s.

c) What is the acceleration experienced by the ball?

a) 13.05 kgm/s b) 6,525 N c) 45,000 m/s2

Example 6.5Example 6.5

The air bag increases the time of the collision; It will also absorb some of the energy from the body It will spread out the area of contact.

A typical 80-kg Physics 231 student (right) drives into a concrete wall at 30 mph = 13.4 m/s. The car buckles by 0.5 m as the car stops.

a) What amount of time does it take to bounce back?(assume constant a)

b) What is the average force exerted on the student?(both in lbs and N)

a) 0.0746 s b) 14,400 N = 31,700 lbs

Elastic & Inelastic CollisionsElastic & Inelastic Collisions

ELASTIC: Both energy & momentum are

conserved

INELASTIC Momentum conserved, not energy Perfectly inelastic -> objects stick Lost energy goes to heat

Examples of Perfectly Inelastic Examples of Perfectly Inelastic CollisionsCollisions

• Catching a baseball• Football tackle• Cars colliding and sticking• Bat eating an insect

Example 6.6Example 6.6

A 5879-lb (2665 kg) Cadillac Escalade going 35 mph =smashes into a 2342-lb (1061 kg) Honda Civic also moving at 35 mph=15.64 m/s in the opposite direction.The cars collide and stick.a) What is the final velocity of the two vehicles?

b) What are the equivalent “brick-wall” speeds for each vehicle?

a) 6.73 m/s = 15.1 mphb) 19.9 mph for Cadillac, 50.1 mph for Civic

Example Example 6.76.7

Ballistic Pendulum: used to measure speed of bullet. 0.5-kg block of wood swings up by height h = 65 cm after stopping 8.0-g bullet.

What was bullet’s velocity? 227 m/s

Example Example 6.86.8

A 5-g bullet traveling at 500 m/s embeds in a 1.495 kg block of wood resting on the edge of a 0.9-m high table. How far does the block land from the edge of the table?

71.4 cm

Example 6.9Example 6.9

Tarzan (M=80 kg) swings on a 12-m vine by letting go from an angle of 60 degrees from the vertical. At the bottom of his swing, he picks up Jane (m=50 kg). To what angle do Tarzan and Jane swing?

35.8 degrees (indepedent of L or g)

Examples of Elastic CollisionsExamples of Elastic Collisions

• pool balls• electron scattering (sometimes)• Earth-superball scattering

Ball Bounce DemoBall Bounce Demo

Example 6.10Example 6.10

An proton (mp=1.67x10-27 kg) elastically collides with a target proton which then moves straight forward. If the initial velocity of the projectile proton is 3.0x106 m/s, and the target proton bounces forward, what are

a) the final velocity of the projectile proton?b) the final velocity of the target proton?

0.03.0x106 m/s

Equal-Mass Collision DemoEqual-Mass Collision Demo

Example 6.11Example 6.11

An proton (mp=1.67x10-27 kg) elastically collides with a target deuteron (mD=2mp) which then moves straight forward. If the initial velocity of the projectile proton is 3.0x106 m/s, and the target proton bounces forward, what are

a) the final velocity of the projectile proton?b) the final velocity of the target deuteron?

vp =-1.0x106 m/svd = 2.0x106 m/s

Head-on collisions with heavier objects always lead to reflections

(Perfectly) Inelastic Collisions(Perfectly) Inelastic Collisions in Two Dimensions in Two Dimensions

fyiyiy

fxixixvmmvmvmvmmvmvm)()(

212211

212211

Two Equations : Two unknowns (vfx, vfy)

Example 6.12Example 6.12

A 1200-kg vehicle moving at 25.0 m/s east collides with a vehicle of mass 1500 kg moving northward at 20.0 m/s. After they join, what is their final speed and direction?

vf = 15.7 m/sf = 45

Elastic Collisions in Two Elastic Collisions in Two DimensionsDimensions

fyfyiyiy

fxfxixix

ffii

vmvmvmvmvmvmvmvm

mvmvmvmv

22112211

22112211

22

21

22

21 2

121

21

21

= 0

Three Equations : Four Unknowns (v1f,v2f,1f,2f)

Example 6.13Example 6.13

A projectile proton moving with v0=120,000 m/s collides elastically with a second proton at rest. If one proton leaves at an angle =30,

a) what is its speed?

b) what are the speed and direction of the second proton? a) v1 = 103,923 m/s

b) v2 = 60,000 m/s 2 = 60

Working out answer in center-of-Working out answer in center-of-massmass

1. Find c.o.m. velocity and subtract it from both v1 and v2

comii

comii

iicom

vvvvvvmm

vmvmv

,2,2

,1,1

21

,22,11

0,22,11 ii vmvm

Note:

2. Problem is easy to solve in this frame:

0:inelasticPerfectly,:elasticPerfectly

21

1211

ff

ffifvv

vvvv

Working out answer in center-of-Working out answer in center-of-massmass

3. Add back c.o.m. velocity to both v1 and v2

comii

comiivvvvvv

,2,2

,1,1

Example 6.14Example 6.14

The mass M1 enters from the left with velocity v0 andstrikes the mass M2=M1 which is initially at rest. The collision is perfectly elastic. ( >, < or =)a) Just after the collision v2 ______ v0.

b) Just after the collision v1 ______ 0.c) At maximum compression, the energy stored in the spring is ________ (1/2)M1v0

2

= =

=

Example 6.15Example 6.15

The mass M1 enters from the left with velocity v0 andstrikes the mass M2<M1 which is initially at rest. The collision is perfectly elastic. ( >, < or =)a) Just after the collision v2 ______ v0.

b) Just after the collision v1 ______ 0.c) Just after the collision p2 ______ M1v0. d) At maximum compression, the energy stored in the spring is ________ (1/2)M1v0

2

> > <

<

Example 6.16Example 6.16

The mass M1 enters from the left with velocity v0 andstrikes the mass M2>M1 which is initially at rest. The collision is perfectly elastic. ( >, < or =)a) Just after the collision v2 ______ v0.

b) Just after the collision v1 ______ 0.c) Just after the collision p2 ______ M1v0. d) At maximum compression, the energy stored in the spring is ________ (1/2)M1v0

2

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