Chapter 9: Linear Momentum and Collisions

Reading assignment: Chapter 10.1-10.5 Homework #16 : (due Monday, Oct. 10, 2005): Problems: Q1, Q14, 9, 14, 21, 28 • Momentum 

p



m v

 • Momentum is conserved – even in collisions with energy loss due to friction/deformation. • Impulse

Black board example 9.3

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. (a) What is the total momentum of the you-and-your-friend system? (b) What is your momentum after you pushed off?

(c) What is your friends speed after you pushed off?

Impulse (change in _________________) A change in _________ is called “impulse”:

J

   

p

 

p f

 

p i

During a collision, a force F acts on an object, thus causing a change in momentum of the object: For a constant (average) force:  

p



J

 

t t i



f

______

dt

 

p



J

 

F 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.

Black board example 10.1

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. (a) What impulse acts on the ball while it is in contact with the foot? (b) The impact time is 0.1s. What is the acting on the ball?

Elastic and inelastic collisions in

one

dimension

________________

is conserved in any collision, elastic and inelastic.

___________________

is only conserved in elastic collisions. Perfectly inelastic collision: After colliding, particles __________________. There

is

a loss of kinetic energy (deformation). Inelastic collisions: Particles _________________ with some loss of kinetic energy.

Perfectly elastic collision: Particles __________________ without loss of kinetic energy.

Perfectly _____________ collision of two particles

(Particles stick together)

m

1

v

 1

i



m

2

v

 2 

p i i

  

p f

(________)



v f K i



E loss



K f

Notice that

p

and

v

are vectors and, thus have a direction (+/-)

1 2

m

1

v

1

i

2 

1 2

m

2

v

2

i

2 

E loss



1 2 (_______)

v f

2 There is a loss in kinetic energy, E loss

Perfectly _________ collision of two particles

(Particles bounce off each other without loss of energy. Momentum is ____________:

m

1

v

 1

i



m

2

v

 2

i



m

1

v

 1

f



m

2 

v

2

f

Energy is _____________ : 1 2

m

1

v

1

i

2  1 2

m

2

v

2

i

2  1 2

m

1

v

1

f

2  1 2

m

2

v

2

f

2

For elastic collisions in : Suppose we know the initial masses and velocities. Then:

v

1

f

  

m

1

m

1 

m

2 

m

2  

v

1

i

  

m

____ 1 

m

2  

v

2

i

(10.38) (10.30)

v

2

f

  

m

____ 1 

m

2  

v

1

i

  

m

2

m

1 

m

1 

m

2  

v

2

i

(10.39) (10.31)

Black board example 9.2

Two carts collide

elastically

on a frictionless track. The first cart (m 1 = 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

inelastic

?

(c) How much energy is lost in the

inelastic

collision?

Black board example 9.5

Ballistic Pendulum:

In a ballistic pendulum a bullet (0.005 kg) is fired into a block (1.0 kg) that is suspended from a light string. The block (with the bullet stuck in it) is lifted up by 0.05 m. (a) What is the speed of the combined bullet/pendulum right after the collision?

(b) Find the initial speed of the bullet?

(c) Find the loss in mechanical energy due to the collision

_______________ collisions

(Two particles) Conservation of momentum:

m

1

v

 1

i



m

2

v

 2

i



m

1 

v

1

f



m

2 

v

2

f

Split into components:

m

1

v

1

ix



m

2

v

2

ix m

1

v

1

iy



m

2

v

2

iy



m

1

v

1

fx



m

1

v

1

fy



m

2

v

2

fx



m

2

v

2

fy

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

Black board example 9.3

Accident 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 Is he telling the truth?

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

How long are the skid marks ( m k = 0.5)