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Math Review Night
Continuous Mass Flow
8.01
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Category 1: Adding Rain
There is a transfer of material into the object but no transfer ofmomentum 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.
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Category 2: Leaking Sand
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.
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Category 3: Impulse
The material continually hits the object providing an impulseresulting 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.
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Category 4: Recoil
The material continually is ejected from the object, resulting in arecoil of the object. For example when fuel is ejected from the back
of a rocket, the rocket recoils forward.
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Worked Example: Coal Car
(constant force, varying speed)An empty coal car of mass m starts from rest under an
applied force of magnitude F. At the same time coal
begins to run into the car at a steady rate b from a coal
hopper at rest along the track. Find the speed when amass mcof coal has been transferred.
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Worked Example: Emptying a
Freight CarAn freight car of mass mccontains a mass of sand ms. At t = 0 a
constant horizontal force of magnitude F is applied in the direction of
rolling and at the same time a port in the bottom is opened to let the
sand flow out at the constant rate b = dms/dt. Find the speed of the
freight car when all the sand is gone. Assume that the freight car is at
rest at t = 0.
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Table Problem: Filling a Cart
Material is blown into cart A from cart B at a rate of b
kilograms per second. The material leaves the chute
vertically downward, so that it has the same horizontal
velocity u, as cart B. At the moment of interest, cart hasmass mAand velocity v. Find an expression for the rate of
change of velocity, the instantaneous acceleration, dv/dt.
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Worked Example: Rocket
A rocket at time t is moving with speed vr,0 in the positive x-direction inempty 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 interms 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
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Strategy : Rocket Problem
Goal: Determine velocity of rocket as function of time asmass is continuously ejected at rate dmf /dt with speed urelative to rocket.
System: consider all elements that undergo momentumchange: rocket and fuel
Using Momentum flow diagram, apply
to find differential equation that describes motion.
total total
ext0
( ) ( )lim
t
t t t
t! "
+ ! #=
!
P PF
! !
!
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Rocket Problem:
A rocket at time t = 0 is moving with speed vr,0in the positive x-
direction in empty space. The rocket burns the fuel at a rate dm f /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.
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State at time t
1.
Rocket with total mass mr(t) moves with speed vr(t) in positive x-direction according to observer
2. Total mass consists of mass of rocket mr,0and fuel mf(t)
3. Fuel element with mass !mf,moves with speed of rocket vr(t) attime t, is ejected during interval [t,t+!t]
4. x-component of momentum at time t
Px ,sys
(t) = (mr(t) + !m
f)v
r(t)
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State at t+ !t
Rocket is propelled forward by ejected fuel with new rocket speed
Fuel is ejected backward with speed u relative to rocket. Relative toobservers frame, ejected fuel element has speed
x-component of systems momentum at time t+!t
vr(t+ !t) = v
r(t) + !v
r
vr
+ !vr" u
P
x ,sys(t+ !t) =m
r(t)(v
r + !v
r) + !m
f(v
r + !v
r" u)
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Rocket EquationAre there any external forces at time t?
Two cases:
(1) Taking off (2) Negligible gravitational field
Apply Momentum Principle:
Conservation of mass: Rate of decrease of mass of rocket equals rate ofejection of mass
ocket equation:
!
Fext
(t) = mr(t)
!
g
!
Fext
=
!
0
total total
ext0
( ) ( )limt
t t t
t! "
+ ! #
=!
P P
F
! !
!
!
Fext
= lim!t"0
mr(t)(
!
vr + !
!
vr) + !m
f
!
u # mr(t)
!
vr
!t= lim!t"0
mr(t)!
!
vr + !m
f
!
u
!t
mr(t)
!
g = mr(t)
d!
vr
dt
+
dmf
dt
!
u
dmf
dt=!
dmr (t)
dt
!
Fext
= mr(t)
d!
vr
dt
!
dmr
dt
!
u
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Rocket Equation in Gravitational
Field Fuel ejection term can be interpreted as thrust force
Relative fuel ejection velocity
External force
Rocket equation
Integrate with respect to time
Solution:
!
Fext+
dmr
dt
!
u = mr(t)
d!
vr
dt
!
u =!u k
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Rocket Equation in Gravitational
Field Fuel ejection term can be interpreted as thrust force
Solution: shorter the burn time, the greater the velocity
!
Fthrust
=
dmr
dt
!
u
!
Fext +
dmr
dt
!
u = mr(t)
d!
vr
dt
vr,z (tf) =u lnmr (t =0)
mr (tf)!gtf
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Concept Question
Suppose rain falls vertically into an open
cart rolling along a straight horizontal track
with negligible friction. As a result of theaccumulating 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.
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Concept Question
If a rocket in gravity-free outer space has the
same thrust at all times, is its acceleration
1.
constant?
2.Increasing?
3.
decreasing?
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Concept Question: Rocket
Fuel Burn TimeWhen a rocket accelerates in a gravitational field, will it reacha greater final velocity if the fuel burn time is
1. as fast as possible?
3. as slow as possible?
5. The final speed is independent of the fuel burn time?
4. Im not sure.