Continuous%20Mass%20Flow%20Rockets%208.01%20W08D1

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Continuous Mass Flow Rockets8.01W08D1
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Juno Lift-Off
http//anon.nasa-global.edgesuite.net/HD_downloads
/juno_launch_1080i.wmv
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Todays Reading Assignment W08D1

• Young and Freedman 8.6
• Class Notes Continuous Mass Flow

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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?

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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.

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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 the
  accumulating water, the speed of the cart
• increases.
• does not change.
• decreases.
• not sure.
• not enough information is given to decide.

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Fire Plane Scooping up Water
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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.
  

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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.

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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.

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Rocket Equation
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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 gt
  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

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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
• to find differential equation that describes
  motion.

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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 gt0.
  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

• 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,0
  and fuel mf (t)
• 3. Fuel element with mass ?mf, moves with speed
  of rocket vr (t) at time t, is ejected during
  interval t, t?t
• 4. x-component of momentum at time 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 to observers frame, ejected
  fuel element has speed
• x-component of systems momentum at time t?t

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Rocket Equation

• Are there any external forces at time t? Two
  cases
• Taking off
• (2) Negligible gravitational field
• Apply Momentum Principle

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Rocket Equation Conservation of Mass and Thrust

• Conservation of mass Rate of decrease of mass
  of rocket equals rate of ejection of mass
• Rocket equation
• Fuel Ejection Thrust force

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Solution Rocket Motion in Gravitational Field

• Rocket Equation
• Fuel ejection term can be interpreted as thrust
  force
• Relative fuel ejection velocity
• External force
• Rocket equation
• Integrate with respect to time
• Solution

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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
• as fast as possible?
• as slow as possible?
• The final speed is independent of the fuel burn
  time?
• Im not sure.

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Answer Rocket Equation in Gravitational Field

• Fuel ejection term can be interpreted as thrust
  force
• Solution shorter the burn time, the greater the
  velocity

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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
• constant?
• increasing?
• decreasing?

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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.
• 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
  
• 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.
• 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.
• 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.

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Next Class W08D2Test Two ReviewCircular Motion,
Energy, Momentum, and Collisions
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Next Reading Assignment W09D1

• Young and Freedman 1.10 (Vector Products)
  9.1-9.6, 10.5