Download Concept Questions

Continuous Mass Flow
Rockets
8.01
W08D1
Juno Lift-Off
http://anon.nasa-global.edgesuite.net/HD_downloads/juno_launch_1080i.wmv
Today’s Reading Assignment:
W08D1
Young and Freedman: 8.6
Class Notes: Continuous Mass Flow
Worked Example: Stream
Bouncing off Wall
A stream of particles of mass m
and separation d hits a
perpendicular surface with
speed v. The stream rebounds
along the original line of motion
with the same speed. The mass
per unit length of the incident
stream is l = m/d. What is the
magnitude of the force on the
surface?
Category 1: Adding Mass But
Not Momentum
There is a transfer of material into the object but no transfer of
momentum in the direction of motion of the object. Consider for
example rain falling vertically downward into a moving cart. A small
amount of rain has no component of momentum in the direction of
motion of the cart.
Concept Question
Suppose rain falls vertically into an open cart
rolling along a straight horizontal track with
negligible friction. As a result of the
accumulating water, the speed of the cart
1.increases.
2.does not change.
3.decreases.
4.not sure.
5.not enough information is given to decide.
Fire Plane Scooping up Water
Category 2: Losing Mass But
Not Momentum
The material continually leaves the object but it does not transport
any momentum away from the object in the direction of motion of the
object. For example, consider an ice skater gliding on ice holding a
bag of sand that is leaking straight down with respect to the moving
skater.
Category 3: Impulse
The material continually hits the object providing an impulse
resulting in a transfer of momentum to the object in the direction of
motion. For example, suppose a fire hose is used to put out a fire on
a boat. The incoming water continually hits the boat impulsing it
forward.
Category 4: Recoil
The material continually is ejected from the object, resulting in a
recoil of the object. For example when fuel is ejected from the back
of a rocket, the rocket recoils forward.
Rocket Equation
Worked Example: Rocket Motion
A rocket at time t is moving with speed vr,0 in the positive x-direction in empty
space. The rocket burns the fuel at a rate dmf,out /dt =b > 0. The fuel is
ejected backward with speed u relative to the rocket.
a) What is the relationship between the time rate of change of exhaust mass
dmf /dt, and the time rate of change of rocket mass dmr /dt?
b) Find an equation for the rate of change of the speed of the rocket in terms
mr (t) ,u, and dmr /dt and solve for v.
c) Find the differential equation describing the motion of the rocket if it is in a
constant gravitational field of magnitude g
Strategy : Rocket Motion
• Goal: Determine velocity of rocket as function of time as
mass is continuously ejected at rate dmf /dt with speed u
relative to rocket.
• System: consider all elements that undergo momentum
change: rocket and fuel
• Using Momentum flow diagram, apply
P total (t  t )  P total (t )
Fext  lim
t 0
t
to find differential equation that describes motion.
Rocket Motion:
A rocket at time t = 0 is moving with speed vr,0 in the positive x-direction in
empty space. The rocket burns the fuel at a rate dmf /dt = b >0. The fuel is
ejected backward with speed u relative to the rocket. The goal is to find
an equation for the rate of change of the speed of the rocket in terms mr (t)
,u, and dmr /dt and solve for v.
State at time t
1.
Rocket with total mass mr(t) moves with speed vr(t) in positive xdirection according to observer
2.
Total mass consists of mass of rocket mr,0 and fuel mf (t)
3.
Fuel element with mass Δmf, moves with speed of rocket vr (t) at
time t, is ejected during interval [t, t+Δt]
4.
x-component of momentum at time t
Px,sys (t)  (mr (t)  m f )vr (t)
State at t + Δt
• Rocket is propelled forward by ejected fuel with new rocket speed
vr (t  t)  vr (t)  vr
• Fuel is ejected backward with speed u relative to rocket. Relative to
observer’s frame, ejected fuel element has speed
vr  vr  u
• x-component of system’s momentum at time t+Δt
Px,sys (t  t)  mr (t)(vr  vr )  m f (vr  vr  u)
Rocket Equation
Are there any external forces at time t? Two cases:
(1)
Taking off
r
r
Fext (t)  mr (t)g
(2) Negligible gravitational field
r
r
Fext  0
Apply Momentum Principle:
P total (t  t )  P total (t )
Fext  lim
t 0
t
r
r
r
r
r
r
mr (t)( v r  v r )  m f u  mr (t)v r
mr (t)v r  m f u
r
Fext  lim
 lim
t0
t0
t
t
r
r
dv r dm f r
Fext  mr (t)

u
dt
dt
Rocket Equation: Conservation of
Mass and Thrust
Conservation of mass: Rate of decrease of mass of
rocket equals rate of ejection of mass
dm f
dmr (t)

dt
dt
r
r
dv r dmr r
Fext  mr (t)

u
Rocket equation:
dt
dt
Fuel Ejection Thrust force:
r
dmr r
Fthrust 
u
dt
r
r
r
r
r
dmr r
dv r
dv r
Fext 
u  mr (t)
 Fext  Fthrust  mr (t)
dt
dt
dt
Solution: Rocket Motion in
Gravitational Field
•
Rocket Equation:
r
r
dmr r
dv r
Fext 
u  mr (t)
dt
dt
r
dmr r
Fthrust 
u
dt
•
Fuel ejection term can be interpreted as thrust force
•
Relative fuel ejection velocity
•
External force
•
•
dvr,z dmr

u
dt
dt
dvr,z
1 dmr
Rocket equation

ug
dt
mr dt
tf
tf
tf
dvr,z
1 dmr
dt   
udt   gdt
Integrate with respect to time t  0 dt
mr dt
t0  0
t0  0
0
•
Solution:
u  uk̂
Fext (t)  mr gk̂
mr g  mr
vr,z (t f )  u ln
mr (t  0)
 gt f
mr (t f )
Concept Question: Rocket
Fuel Burn Time
When a rocket accelerates in a constant gravitational
field, will it reach a greater final velocity if the fuel burn
time is
1. as fast as possible?
2. as slow as possible?
3. The final speed is independent of the fuel burn
time?
4. I’m not sure.
Answer: Rocket Equation in
Gravitational Field
• Fuel ejection term can be interpreted as thrust force
r
r
dmr r
dv r
Fext 
u  mr (t)
dt
dt
r
dmr r
Fthrust 
u
dt
• Solution: shorter the burn time, the greater the velocity
mr (t  0)
vr,z (t f )  u ln
 gt f
mr (t f )
Concept Question: Rocket with
Constant Thrust
If a rocket in gravity-free outer space has a constant
thrust at all times while burning fuel, is its
acceleration
1. constant?
2. increasing?
3. decreasing?
Table Problem: Rocket Sled
A rocket sled ejects gas backwards at a speed u relative to the rocket sled.
The mass of the fuel in the rocket sled is equal to one half the initial total mass mr,0
(including fuel) of the sled. The rocket sled starts from rest on a frictionless track.
You may ignore air resistance.
a) Derive a relation between the differential of the speed of the rocket sled, dvr , and
the differential of the total mass of the rocket, dmr
b) Integrate the above relation to find the speed of the rocket sled as a function of
mass, vr(m), as the rocket sled speeds up.
c) What is the final speed of the rocket sled after all the fuel has been burned?
Express your answers in terms of the quantities u, and mr,0 as needed.
d) After reaching its final speed, the sled enters a rough portion of the track that
begins at x = 0 with a coefficient of kinetic friction that varies with distance
μk(x)=bx where b is a positive constant. How far D did the sled slide before it came
to rest in that portion of the track? Express your answers in terms of the quantities
u, b, g, and mr,0 as needed.
Next Class W08D2
Test Two Review
Circular Motion, Energy,
Momentum, and Collisions
Next Reading Assignment:
W09D1
Young and Freedman: 1.10 (Vector
Products) 9.1-9.6, 10.5