Continuous Mass Flow Rockets 8 01 W05D2 Today s Reading MIT 8 01 Course Notes Chapter 11 Reference Frames Sections 11 1 11 3 Chapter 12 Momentum and the Flow of Mass Sections 12 1 12 3 Announcements Problem Set 4 due Week 6 Tuesday at 9 pm in box outside 26 152 Math Review Week 6 Tuesday 9 11 pm in 26 152 Add Date Friday Oct 4 Conservation of Momentum System For a fixed choice of system if there are no external forces acting on the system then the momentum of the system is constant is constant Fext 0 psystem 0 Strategy Momentum of a System 1 Choose system 2 Identify initial and final states 3 Identify any external forces in order to determine whether any component of the momentum of the system is constant or not i If there is a non zero total external force dp sys total Fext dt ii If the total external force is zero then momentum is constant p sys 0 p sys f External Forces and Constancy of Momentum Vector The external force may be zero in one direction but not others The component of the system momentum is constant in the direction that the external force is zero The component of system momentum is not constant in a direction in which external force is not zero Modeling Instantaneous Interactions Decide whether or not an interaction is instantaneous External impulse changes the momentum of the system t tcol I t t tcol Fext dt Fext ave tcol p sys t If the collision time is approximately zero tcol 0 then the change in momentum is approximately zero p system 0 Problem Solving Strategy Momentum Diagrams 1 Identify all objects in system 2 Choose a reference frame 3 Draw a momentum diagram for i state of the system before interaction ii state of system after collision Diagram should include 1 Choice of unit vectors for direction of momentum 2 Show each mass element Show the speed and arrow for direction of velocity of each mass element Momentum and External Forces Momentum law Fext dPsys dt For discrete changes Fext ave tinteraction Psys f t f Psys i ti For continuous changes Fext lim t 0 Psys t t Psys t t Recoil Relative Speed The person jumps off the cart with a speed u relative to the final speed of the cart vc The speed of the person vp relative to ground is v p u vc Table Problem Recoil A person of mass mp is standing on a cart of mass mc that is on ice Assume that the contact between the cart s wheels and the ice is frictionless The person jumps off the cart with speed u relative to the final speed of the cart a Does the momentum of the system person and cart change when the person jumps off a What is the final velocity of the cart as seen by an observer fixed to the ground b What is the final velocity of the person as seen by an observer fixed to the ground Continuous Mass Transfer Concept Question Rain Falling Into Cart 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 Concept Question Losing Mass But Not Momentum Consider an ice skater gliding on ice holding a bag of sand that is leaking straight down with respect to the moving skater As a result of the leaking sand the speed of the skater 1 increases 2 does not change 3 decreases 4 not sure Continuous 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 Mini Experiment Recoiling Balloons Rocket Equation Consider the propulsion as a series of recoils Rocket Motion in Empty Space A rocket at time t 0 is initially at rest in empty space Fuel is ejected continually backward with speed u relative to the rocket during the interval 0 t We shall divide up this time interval into a sequence of N time intervals in which the amount of fuel m f is ejected in each interval 0 t1 t1 t1 t2 t1 t2 t N 1 t N t N 1 t N Rocket Motion in Empty Space For each interval we will calculate the change in speed of the rocket due to recoil For interval 0 t1 t1 calculate vr 1 For interval t1 t2 t1 t2 calculate vr 2 For interval t N 1 t N t N 1 t N t calculate vr N Rocket Motion in Empty Space To find the final speed of the rocket we then sum up the changes in speed for each interval j N vr t vr j j 1 Now we take the limit that vr j 0 N Our sum then goes to an integral vr t vr t vr 0 0 dvr Recoil During Interval 0 t1 In the first interval recoil of rocket can be calculated as follows px i px f 0 m f u vr 1 mr 1 vr 1 m f u m f vr 1 mr 1 vr 1 Recoil During Interval 0 t1 The speed of the ejected fuel is much greater than the small recoil speed of the rocket and therefore m f u m f vr 1 So our momentum principle enables us to calculate the recoil speed for the first interval m f u mr 1 vr 1 vr 1 In the limit that m f mr 1 u t1 0 vr 1 0 and even for small u m f u m f vr 1 Recoil During Interval t1 t2 vr 2 m f mr 2 u In the second interval we choose a new reference frame S1 that is moving with speed vr 1 with respect to our first reference S0 Then the rocket is again at rest at t1 in this new reference frame and so our calculation for the second recoil is identical to our previous case Recoil During Interval tj 1 tj For an arbitrary interval the same argument applies where S j 1 is the reference frame which is moving with speed vr j 1 vr 1 vr 1 vr j 1 vr j m f mr j u with respect to S0 The rocket is at t j 1 rest in this frame at time Recoil During Interval tj 1 tj For an arbitrary interval same argument applies where S j 1is the reference frame which is moving with speed vr j 1 vr 1 vr 2 vr j 1 vr j m f mr j u with respect to S0 The rocket is at rest in this frame at time t j 1 Recoil Contributions to Speed of Rocket The speed at time t j is the sum of contributions vr j vr 1 vr 2 vr j vr j m f mr 1 u m f mr …
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