Math Review Class Continuous Mass Flow

8/11/2019 Math Review Class Continuous Mass Flow

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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.


8/20

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


17/20

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


18/20

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.

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Math Review Class Continuous Mass Flow

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