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Transcript
Lecture #2
Questions on Assignment #1?
Homework Expectations /Office hours
Impulse and momentum conservation
Rocket propulsion
Worked problems
Cross-product problem
Impulse and Momentum V-5-7
1 :10
Homework Bliss
GOOD
NOT GOOD
mr  mass of cannon  2.1103 kg
m1  2.1103
mc  mass of chicken  2.4 kg
m2  2.4
J  cannon impulse  (1000 Nt  s ) xˆ
impulse  1000
vic , v fc  initial , final velocity of chicken
vi , v f
mcannon  2.110 kg ; mchicken  2.4kg
3
vi chicken , v f chicken
Brief description
# of
points
No attempt
Just started /
confused
Well along
Nearly perfect
Physics Bliss
0
4
7
9
10
2 :17
Definition of Impulse
J  F (t ) t
J 

t2
F (t ) dt
Impulse J is a useful concept in
the study of collisions.
(e.g. Balls and bats, automobiles,
comets and planets)
Impulse is the average force acting
W  F ( r ) r
over a time period multiplied by
r2
the time period.
W 
F ( r ) dr
r1
It may also be written as an integral
Note the difference between impulse J
and work W.
t1

3 :22
Definition of Impulse
J  F (t ) AV t
t2
J   F (t )dt
t1
W  FAV s
r2
W  F d
r1
Impulse J is a useful concept in
the study of collisions.
(e.g. Balls and bats, automobiles,
comets and planets)
Impulse is the average force acting
over a time period multiplied by
the time period.
It may also be written as an integral
Note the difference between impulse J
and work W.
4 :22
Impulse and momentum change
t2
J   F (t )dt
t1
J 
t2
t1
dp(t )
dt  p (t2 )  p (t1 )  p
dt
Two equivalent impulses with different Fmax
1
Force (N)
0.8
0.6
0.4
0.2
0
0
2
4
6
Time (s)
8
10
Impulse is useful
because it
directly
allows
expression of
momentum
change.
5 :27
Impulse I -- Problem #L2-2
“A car crash”
Jack and Jill were ingesting things they shouldn’t
while driving two cars of mass 1000
 kg and
2000 kg with velocity
vectors 30 x m/s

and 10 x  60 y m/s
We may look at their collision in terms of impulse.
Jill’s car applied an impulse to Jack’s car as
follows: J  (13.33x
ˆ  40 yˆ ) 103 Nt  s
What is the final momentum of Jack’s car
only? What is his final velocity?
Calculate
pJack  final , vJack  final and vJack  final
6 :32
Rocket Science I


F  ma
 Not the whole story


dP
 More general – get used to it
F 
dt
 Case where m not constant
 d
F  mv
dt
 v)
 (mv  m
 vexhaust defined as " thrust"
m

marocket  thrust  Instantaneous acceleration, but
rocket keeps getting lighter!!
7 :40
Rocket Science II

mr v  ve m
In absence of other
forces
v
m f dm
dv  ve
Separable ODE.
0
mi
m
Integrate both
sides
 mf 

v f  ve ln 

Rocket
velocity
in
 mi 
terms of lost mass
 m f  and exhaust
 velocity
v f  2.3ve log 10 


 mi 
8 :45
Rocket Science III
 mf
v f  2.3ve log 10 
 mi



If a rocket is
60% fuel, then
it limits at ve
If a rocket is
90% fuel, then
limit at 2.3 ve
At 99% fuel,
limit is 4.6 ve
Achieving escape velocity with Single-Stage to Orbit rocket
70000
Exhaust velocity 3350 mph
Exhaust velocity 6700 mph
Exhaust velocity 13400
Escape velocity
65000
60000
55000
50000
Velocity (mph)
45000
40000
35000
30000
25000
20000
15000
10000
5000
0
90
80
70
60
50
40
30
Percentage of rocket mass remaining
20
10
1
9 :50
Specific Impulse
I sp
Engine Type
Best Chemical
Best Nuclear
Xenon Ion
Linear Accel.
EM Catapult
ve

g
Specific
Impulse
500
2,000
3,800
1,000,000
N/A
 In practice of rocketry, the “specific
impulse” is often quoted
Exhaust Velocity
m/s
mph
4,900
10,363
19,600
41,452
37,240
78,759
9,800,000
20,726,119
45,000,000
95,170,955
Rocket
Vfinal
mph
23,835
95,340
181,146
47,670,074
95,170,955
Travel to
Alpha-C
(yrs)
106,477
26,619
14,010
53
27
10 :45
:55
XIPS
 Xenon Ion Propulsion System
 Thrusts are 60-200 milliNewtons
 Used for “station-keeping” and deepspace missions
11 :45
:55
Problem T3.7
The first couple of minutes after a space shuttle
launch can be described as follows: The
initial mass is 2x10E6 kg, the final mass (after
2 minutes) is about 1x10E6 kg, the average
exhaust speed is about 3000 m/s. If all this
were taking place in outer space, with
negligible gravity, what would be the shuttle’s
speed at the end of this stage? What is the
thrust during this same period and how does
it compare with the total initial weight of the
shuttle (on earth)?
12 :65
Impulse II -- Problem #L2-3
“Another car crash”
Jack and Jill are partly recovered from their previous
injuries, and haven’t learned from their experience.
They are drinking cokes with Bacardi 151 and not
wearing seat-belts.
Jack’s vehicle has velocity vector 30 xˆ m / s
30 xˆ m / s
Jill’s vehicle has
Both vehicles’ mass=M. Both people’s mass=70 kg.
Solve for case of inelastic and elastic collisions of vehicles.
Jill has an airbag in her vehicle. It takes her 100 millisec to
reduce her velocity relative to her vehicle to zero.
Jack stops 5 millisec after impacting the steering wheel.
What impulse does each person experience? What is
13
average force for each? How many “g’s” do they feel.:75
Lecture #2 Wind-up
. Impulse J  t F (t )dt  p
t
. mv
exhaust defined as " thrust "
.
 mf 
v f  ve ln 

 mi 
Got on WebCT? / Got Books?
Office hours Wednesday 4-5:30
Homework problems in Taylor, Handout,
Galileo problem – check the web.
Second homework due in class Thursday 9/4
2
1

(Includes introducing gravity into rocket equation)
14 :72