momentum momentum is conserved – even in collisions with energy loss. collisions

• Momentum • Momentum is conserved – even in collisions with energy

loss. • Collisions• Center of mass• Impulse

Chapter 9: Linear Momentum and Collisions

Reading assignment: Chapter 9.1 to 9.6

Homework: QQ2, AE1, AE5, 5, 9, 10, 11, 14, 19, 27, 28, 29, 32, 33, 37, 38, 65

Due date: Monday, March 21

vmp

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The linear momentum of a particle of mass m and velocity v is defined as

vmp

The linear momentum is a vector quantity.

It’s direction is along v.

zzyyxx vmpvmpvmp The components of the momentum of a particle:

Chapter 9: Linear Momentum and Collisions

1, 2, 1, 2,

constant

or:

i f

i i f f

p p

p p

p p p p

Conservation of linear momentum

Black board example 9.1(similar to blocks and spring HW problem)

You (100kg) and your skinny friend (50.0 kg) stand face-to-face on a frictionless, frozen pond. You push off each other. You move backwards with a speed of 5.00 m/s.

Quick quiz and demo:

How are rocket ships (in space) able to change their velocity?

2. What is your momentum after you pushed off?

A. 0 kgm/sB. 250 kgm/sC. 500 kgm/sD. 750 kgm/sE. -500 kgm/s

1. What is the total momentum of the you-and-your-friend system?

A. 0 kgm/sB. 250 kgm/sC. 500 kgm/sD. 750 kgm/sE. -500 kgm/s

4. How much energy (work) did you and your friend expend?

3. What is your friends speed after you pushed off?A. 0 m/sB. 5 m/sC. 10 m/sD. -5 m/sE. -10m/s

Elastic and inelastic collisions in one dimension

Momentum is conserved in any collision, elastic and inelastic.

Mechanical Energy is only conserved in elastic collisions.

Perfectly inelastic collision: After colliding, particles stick together. There is a loss of energy (deformation).

Elastic collision: Particles bounce off each other without loss of energy.

Inelastic collision: Particles collide with some loss of energy, but don’t stick together.

Perfectly inelastic collision of two particles(Particles stick together)

fii

fi

vmmvmvm

pp

)( 212211

Notice that p and v are vectors and, thus have a direction (+/-)

lossfii

flossi

Evmmvmvm

KEK

221

222

211 )(

21

21

21

There is a loss in energy, Eloss

Perfectly elastic collision of two particles(Particles bounce off each other without loss of energy.

ffii vmvmvmvm 22112211

222

211

222

211 2

121

21

21

ffii vmvmvmvm

Energy is conserved:

Momentum is conserved:

1 1 2 2i f i fv v v v

By plugging one equation into the other, we can also derive:

Black board example 9.2

Two carts collide elastically on a frictionless track. The first cart (m1 = 1kg) has a velocity in the positive x-direction of 2 m/s; the other cart (m = 0.5 kg) has velocity in the negative x-direction of 5 m/s.

(a) Find the speed of both carts after the collision.

(b) What is the speed if the collision is perfectly inelastic?

(c) How much energy is lost in the inelastic collision?

Black board example 9.3 and demoDetermining the speed of a bullet

A bullet (m = 0.01kg) is fired into a block (0.1 kg) sitting at the edge of a table. The block (with the embedded bullet) flies off the table (h = 1.2 m) and lands on the floor 2 m away from the edge of the table.

a.) What was the speed of the bullet?b.) What was the energy loss in the bullet-block collision?

vb = ?

h = 1.2 m

x = 2 m

Two-dimensional collisions (Two particles)

1 1 2 2 1 1 2 2

i f

i i f f

p p

m v m v m v m v

Conservation of momentum:

Split into components:

, ,

1 1 2 2 1 1 2 2

, ,

1 1 2 2 1 1 2 2

x i x f

ix ix fx fx

y i y f

iy iy fy fy

p p

m v m v m v m v

p p

m v m v m v m v

If the collision is elastic, we can also use conservation of energy.

Black board example 9.4Accident investigation. Two automobiles of equal mass approach an intersection. One vehicle is traveling towards the east with 29 mi/h (13.0 m/s) and the other is traveling north with unknown speed. The vehicles collide in the intersection and stick together, leaving skid marks at an angle of 55º north of east. The second driver claims he was driving below the speed limit of 35 mi/h (15.6 m/s).

13.0 m/s

??? m/s

a) Is he telling the truth?

b) What is the speed of the “combined vehicles” right after the collision?

c) How long are the skid marks (mk = 0.5)?

Motion of a System of Particles.

Newton’s second law for a System of Particles

The center of mass of a system of particles (combined mass M) moves like one equivalent particle of mass M would move under the influence of an external force.

zCMznetyCMynetxCMxnet

CMnet

MaFMaFMaFaMF

,,,,,,

Center of mass

Center of mass for many particles:M

rmr i

ii

CM

Where is the center of mass of this arrangement of particles.

(m3 = 2 kg; m1 = m2 = 1 kg)?

M

ama i

ii

CM

M

vmv i

ii

CM

Velocity of the center of mass: Acceleration of the center of mass:

Black board example 9.5

A rocket is shot up in the air and explodes.

Describe the motion of the center of mass before and after the explosion.

A method for finding the center of mass of any object.

- Hang object from two or more points.

- Draw extension of suspension line.

- Center of mass is at intercept of these lines.

Impulse (change in momentum)

A change in momentum is called “impulse”: if pppJ

During a collision, a force F acts on an object, thus causing a change in momentum of the object:

f

i

t

t

dttFJp )(

For a constant (average) force: tFJp avg

Think of hitting a soccer ball: A force F acting over a time t causes a change p in the momentum (velocity) of the ball.

A soccer player hits a ball (mass m = 440 g) coming at him with a velocity of 20 m/s. After it was hit, the ball travels in the opposite direction with a velocity of 30 m/s.

Black board example 9.6

1. What impulse acts on the ball while it is in contact with the foot?

2. The impact time is 0.1s. What average force is the acting on the ball?

3. How much work was done by the foot? (Assume an elastic collision.)

1A. 0

1B. 20 kg m/s‧

1C. 22 kg m/s‧

1D. 30 kg m/s‧

1E. 33 kg m/s‧

2A. 0

2B. 200 N

2C. 220 N

2D. 300 N

2E. 330 N

3A. 0

3B. 110 J

3C. 220 J

3D. 300 J

3E. 330 J