Physics 121,Lecture 15,Pg 1
Physics 121,Sections 9,10,11,and 12
Lecture 15
Today’s Topics:
? Homework 6,
? Chap,6,# 6,12,20,24,29,38,52,57,78,and 83.
? Midterm 1:
? Average,65%
? Chapter 6:
? Escape velocity
? Chapter 7:
? Linear momentum
? Collision
M i d t e rm 1
0
2
4
6
8
10
12
14
16
5
10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95
1
0
0
M
o
re
S c o r e
N
u
mb
e
r
o
f
s
t
u
d
e
n
t
s
Physics 121,Lecture 15,Pg 2
Problem,How High?
? A projectile of mass m is launched straight up from
the surface of the earth with initial speed v0,What is
the maximum distance from the center of the earth
RMAX it reaches before falling back down.
RMAX
RE v
0
m
M
Physics 121,Lecture 15,Pg 3
Problem,How High...
? No non-conservative
forces:
?WNC = 0
??K = -?U
RMAX
v0
m
hMAX
? And we know:
??
?K ? ? 12 mv 02
??
? ? U ? ? G M m 1R
E
? 1R
M A X
????
??
????
??
RE
M ?
????? ??
M A XE
2
0 R
1
R
1G M mmv
2
1
Physics 121,Lecture 15,Pg 4
E
2
0
M A X
E
M A X
E
E
M A X
E
E2
E
M A XE
2
0
M A XE
2
0
gR2
v
R
R
1
R
R
1gR2
R
R
1R
R
GM
2
R
1
R
1
GM2v
R
1
R
1
G M mmv
2
1
??
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RMAX
RE v
0
m
hMAX
MR
R
v
gR
M AX
E
E
?
?1
2
0
2
Problem,How High...
Physics 121,Lecture 15,Pg 5
Escape Speed (Velocity)
? If we want the projectile to escape to infinity we need
to make the denominator in the above equation zero:
R
R
v
gR
M AX
E
E
?
?1
2
0
2
1
2
00
2
? ?v
gR E
v
gR E
0
2
2 1?v gR E0 2?
We call this value of v0 the escape velocity vesc
Physics 121,Lecture 15,Pg 6
Escape Velocity
? Remembering that we find the escape velocity
from a planet of mass Mp and radius Rp to be:
(where G = 6.67 x 10-11 m3 kg-1 s-2).
v
GM
Resc
p
p
? 2
g GM
R E
? 2
Moon
Earth
Sun
Jupiter
Rp(m) Mp(kg) gp(m/s2) vesc(m/s)
6.378x106 5.977x1024
1.738x106 7.352x1022
7.150x107 1.900x1027
6.960x108 1.989x1029
9.81
1.62
24.8
27.4
11.2x103
2.38x103
59.5x103
195.x103
Physics 121,Lecture 15,Pg 7
Lecture 15,Act 1
Escape Velocity
? Two identical spaceships are awaiting launch on two planets
with the same mass,Planet 1 is stationery,while Planet 2 is
rotating with an angular velocity ?.
?Which spaceship needs more fuel to escape to infinity.
(a) 1 (b) 2 (c) same
(1) (2)
?
Physics 121,Lecture 15,Pg 8
Lecture 15,Act 1
Solution
? Both spaceships require the same escape velocity
to reach infinity,
? Thus they require the same kinetic energy,
? Both initially have the same potential energy.
? Spaceship 2 already has some kinetic energy due
to its rotational motion,so it requires less work (i.e,
less fuel).
Physics 121,Lecture 15,Pg 9
Aside
? This is one of the reasons why all of the world’s
spaceports are located as close to the equator as
possible.
r1
r2
r2 > r1
K2 = m(?r2)2 >21 K1 = m(?r1)221
v = ?r
Physics 121,Lecture 15,Pg 10
Algebraic Solution
WNC = ?K + ?U = ?E
For spaceship 1,W1 = (Kf - K0) + (Uf - U0)
K0 = 0,U0 = 0 W1 = Kf + Uf
For spaceship 2,W2 = (Kf - K0) + (Uf - U0)
K0 = m(?r)2,U0 = 0 W2 = Kf + Uf - m(?r)2
W1 > W2 So spaceship 1 will need more fuel,
Physics 121,Lecture 15,Pg 11
Chapter 7
Linear Momentum
? Units of linear momentum are kg m/s.
p = mv (p is a vector since v is a vector).
So px = mvx etc.
? Definition,For a single particle,the linear momentum p is
defined as:
? Newton’s 2nd Law:
Physics 121,Lecture 15,Pg 12
Impulse-momentum theorem:
? The impulse of the force action on an object equals the change
in momentum of the object:
F?t = ?p = mvf – mvi
Impulse? F?t = ?p
? The impulse imparted by a force during the time interval ?t is
equal to area under the force-time graph from beginning to the
end of the time interval.
F
t
ti tf
?t
Fav?t
F
t
ti tf
?t
Impulse has units of Ns.
Physics 121,Lecture 15,Pg 13
Average Force and Impulse
?t
F
t
F
t
?t
?t big,Fav small
?t small,Fav big
soft spring
stiff spring
Fav
Fav
Physics 121,Lecture 15,Pg 14
Momentum 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 energy is not conserved.
t
p
E X T ?
??F 0???tp 0?EXTF
Physics 121,Lecture 15,Pg 15
Elastic 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.
vi
? 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.
Physics 121,Lecture 15,Pg 16
Inelastic 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
Physics 121,Lecture 15,Pg 17
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 !
?Px,before = Px,after
?mv = (M+m) V
v V
before after
v M mm V? ???? ???
x
Physics 121,Lecture 15,Pg 18
Example 1...
? Now consider the energy of the system before and after:
? Before:
? After:
? So
? ?E mv m M m
m
V M m
m
M m VB ? ? ???? ??? ? ???? ??? ?1
2
1
2
1
2
2
2
2 2
? ?E M m VA ? ?12 2
E mM m EA B? ???? ???
Energy is NOT conserved (friction stopped the bullet)
However momentum was conserved,and this was useful.
v M mm V? ???? ???
Physics 121,Lecture 15,Pg 19
Lecture 15,Act 2
Inelastic Collision in 1-D
ice
(no friction)
Winter in Storrs
Physics 121,Lecture 15,Pg 20
Lecture 15,Act 2
Inelastic Collision in 1-D
vf =?
finally
m
v = 0 ice
M = 2m
V0 (no friction)
initially
Vf = A) 0 B) Vo/2 C) 2Vo/3 D) 3Vo/2 E) 2Vo
Physics 121,Lecture 15,Pg 21
Lecture 15,Act 2
Inelastic Collision in 1-D
Use conservation of momentum to find v after the collision.
Before the collision:
)0(mMi ?? oVP
After the collision:
ff mM VP )( ??
ooo VVVV 3
2
)2(
2
)( ????? mm
m
mM
M
f
Conservation of momentum,fi PP ? fmMM VV o )( ??
C) 2Vo/3
Physics 121,Lecture 15,Pg 22
Lecture 15,ACT 3
Momentum 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) Box 1 (b) Box 2 (c) same
1 2
Physics 121,Lecture 15,Pg 23
Lecture 15,ACT 3
Momentum Conservation
? 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 !
1 2
x
V1 V2
Physics 121,Lecture 15,Pg 24
Lecture 15,ACT 3
Momentum Conservation
1 2
x
V1 V2
mv = MV1 - mv’
V1 = (mv + mv’) / M
mv = (M+m)V2
V2 = mv / (M+m)
V1 numerator is bigger and its denominator is smaller
than that of V2.
V1 > V2
Physics 121,Lecture 15,Pg 25
Inelastic collision in 2-D
? Consider a collision in 2-D (cars crashing at a slippery
intersection...no friction).
v1
v2
V
before after
m1
m2
m1 + m2
Physics 121,Lecture 15,Pg 26
Inelastic collision in 2-D...
? There are no net external forces acting.
?Use momentum conservation for both components.
v1
v2
V = (Vx,Vy)
m1
m2
m1 + m2
b,xa,x PP ? ? ?m v m m V x1 1 1 2? ?? ?
V mm m vx ? ? 1
1 2
1
b,ya,y PP ? ? ?m v m m V y2 2 1 2? ?? ?
V mm m vy ? ? 2
1 2
2
Physics 121,Lecture 15,Pg 27
Inelastic collision in 2-D...
? So we know all about the motion after the collision !
V = (Vx,Vy)
Vx
Vy
?
? ?V
m
m m vx ? ?
1
1 2
1
? ?V
m
m m vy ? ?
2
1 2
2
1
2
11
22
x
y
p
p
vm
vm
V
Vt a n ????
Physics 121,Lecture 15,Pg 28
Inelastic collision in 2-D...
? We can see the same thing using vectors:
tan ? ? pp 2
1
P
p1
p2
P
p1
p2
?
Physics 121,Lecture 15,Pg 29
Explosion (inelastic un-collision)
Before the explosion,M
m1 m2
v1 v2
After the explosion:
Physics 121,Lecture 15,Pg 30
Explosion...
? No external forces,so P is conserved.
? Initially,P = 0
? Finally,P = m1v1 + m2v2 = 0
m1v1 = - m2v2 M
m1 m2
v1 v2
Physics 121,Lecture 15,Pg 31
Lecture 15,ACT 4, Center of Mass
? A bomb explodes into 3 identical pieces,Which of the following
configurations of velocities is possible?
(a) 1 (b) 2 (c) both
m m
v V
v
m
m m
v v
v
m
(1) (2)
Physics 121,Lecture 15,Pg 32
Lecture 15,ACT 4, Solution
m m
v v
v
m
(1)
? No external forces,so P must be conserved.
? Initially,P = 0
? In explosion (1) there is nothing to balance the upward
momentum of the top piece so Pfinal ? 0.
mvmv
mv
Physics 121,Lecture 15,Pg 33
? No external forces,so P must be conserved.
? All the momenta cancel out.
? Pfinal = 0.
(2)
m m
v v
v
m
mv
mv
mv
Lecture 15,ACT 4, Solution
Physics 121,Lecture 15,Pg 34
Comment 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.
Physics 121,Lecture 15,Pg 35
Ballistic Pendulum
? 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?
H
L L
L L
m
M
M + mv
V
V=0
Physics 121,Lecture 15,Pg 36
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 energy E.
(no non-conservative forces acting after collision)
Physics 121,Lecture 15,Pg 37
Ballistic Pendulum...
? Stage 1,Momentum is conserved
in x-direction,mv m M V? ?( )
V mm M v? ???? ???
? Stage 2,Energy is conserved
( )E EI F?
12 2( ) ( )m M V m M gH? ? ?V gH2 2?
Eliminating V gives,gH2m
M1v ?
?
??
?
? ??
Physics 121,Lecture 15,Pg 38
Ballistic Pendulum
? If we measure the forward displacement d,not H:
H
L L
L L
m
M
M + mv
d
L
d H
L-H
? ?L d L H
H L L d
2 2 2
2 2
? ? ?
? ? ?
Physics 121,Lecture 15,Pg 39
Ballistic Pendulum
L
d H
L-H
L2
d
L2
d1LL
L
d1LL
dLLH
22
2
2
22
???
?
?
???
? ??????
???
d
L ?? 1for
gH2mM1v ?????? ?? LgdmM1v ???????? ??
for d << L
Physics 121,Lecture 15,Pg 40
Recap of today’s lecture
? Homework 6,
?Chap,6,# 6,12,20,24,29,38,52,57,78,and 83.
? Midterm 1:
?Average,65%
? Chapter 6:
?Escape velocity
? Chapter 7:
?Linear momentum
?Collision