Physics 2211, Spring 2002

1
Physics 2211 Lecture 22Todays Agenda

• Impulse
• Conservation of momentum
• Inelastic collisions in one dimension
• Elastic collisions in one dimension

2
Collision time scales

• Collisions typically involve interactions that
  happen quickly.

3
Collision time scales

• During this brief time, the forces involved can
  be quite large

?t
t1
t5
t2
t4
t3
p1
p2
p3 0
p5
p4
F2
F4
F3
4
Force and Impulse
5
Force and Impulse
6
Force and Impulse

• Two different collisions can have the same
  impulse since dependsonly on the change in
  momentum,not the nature of the collision.

same area
F
t
?t
?t
ti
tf
ti
tf
?t big, F small
?t small, F big
7
Force and Impulse
soft spring
F
stiff spring
t
?t
?t
ti
tf
ti
tf
?t big, F small
?t small, F big
8
Force and Impulse

• We can use the notion of impulse to define
  average force, which is a useful concept.

The time average of a force for the time interval
?t tf - ti is
9
Force and Impulse
soft spring
Fav
F
stiff spring
Fav
t
?t
?t
ti
tf
ti
tf
?t big, Fav small
?t small, Fav big
10
Force and ImpulseBaseball Example

• A pitcher pitches the ball (m .7 kg) at 145
  km/hr (about 90 mph).
• The batter makes contact with the ball for .001 s
  causing the ball to leave the bat going 190 km/hr
  (about 120 mph).
• Find the average force on the ball, disregarding
  gravity.

11
Baseball Example
12
Conservation of Linear MomentumReview

• Conservation of Linear Momentum

13
Comment on Energy Conservation

• Total kinetic energy of a system undergoing an
  inelastic collision is not conserved.
• Energy is lost
• Heat (bullet in block)
• 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, momentum conservation is easier to
  satisfy than energy conservation.

14
Conservation of Momentum Example

• 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 (totally inelastic
  collision).
• Which box ends up moving faster?

15
Conservation of Momentum Example

• Since the total external force in the x-direction
  is zero, momentum is conserved along the x-axis.
• In both cases the initial momentum is the same
  (mv of ball).
• In case 1 the ball has negative momentum after
  the collision, hence the box must have more
  positive momentum if the total is to be
  conserved.
• The speed of the box in case 1 is biggest!

x
V1
V2
2
1
16
Conservation of Momentum Example
17
Conservation of Momentum Example

• Is case1 an elastic collision?

18
Ballistic Pendulum
L
L
V 0
L
L
H
m
v
M m
V
M

• A projectile of mass m moving horizontally with
  speed v strikes a stationary mass M suspended by
  strings of length L. Subsequently, m M rise
  to a height of H.

Given H, what is the initial speed v of the
projectile?
19
Ballistic Pendulum

• Two stage process

1. m collides with M, inelastically. Both M and
m then move together with a velocity V (before
having risen significantly).
2. M and m rise a height H, conserving K U
energy E. (no non-conservative forces acting
after collision)
20
Ballistic Pendulum

• Stage 1 Momentum is conserved

in x-direction
21
Elastic Collisions

• Elastic means that kinetic energy is conserved as
  well as momentum.
• This gives us more constraints
• We can solve more complicated problems!!
• Billiards (2-D collision)
• The colliding objectshave separate motionsafter
  the collision as well as before.
• Start with a simpler 1-D problem

Initial
Final
22
Elastic Collision in 1-D
m2
m1
initial
v1i
v2i
x
23
Elastic Collision in 1-D
Conserve PX
Conserve Kinetic Energy
The rate of approach rate of recession
24
Basketball Demo

• Carefully place a small rubber ball (mass m) on
  top of a much bigger basketball (mass M). Drop
  these from some height. The height reached by the
  small ball after they bounce is 9 times the
  original height!! (Assumes M gtgt m and all bounces
  are elastic).
• Understand this using the speed of approach
  speed of recession property we just proved.

3v
m
v
v
v
v
M
v
(a)
(b)
(c)
25
Basketball Demo
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Recap of todays lecture

• Impulse
• Conservation of momentum
• Inelastic collisions in one dimension
• Elastic collisions in one dimension
• Review Section 8.6 inTipler