W8.4 - February 28, 2008 Physics for...

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Unformatted text preview: February 28, 2008 Physics for Scientists&Engineers 1 1 Physics for Scientists & Physics for Scientists & Engineers 1 Engineers 1 Lecture 30 February 28, 2008 Physics for Scientists&Engineers 1 2 Rocket Motion Rocket Motion Here we consider the case where the object in motion changes mass For example, the space shuttle in flight burning its fuel A rocket works by ejecting material This material is usually the result of a chemical reaction February 28, 2008 Physics for Scientists&Engineers 1 2 Rocket Motion Rocket Motion Here we consider the case where the object in motion changes mass For example, the space shuttle in flight burning its fuel A rocket works by ejecting material This material is usually the result of a chemical reaction February 28, 2008 Physics for Scientists&Engineers 1 3 Toy Model of a Rocket Toy Model of a Rocket Lets imagine a spaceship that powers itself by shooting cannonballs out its nozzle Each cannonball has mass m and the initial mass of the rocket is m Each cannonball is fired with velocity v c relative to the rocket producing a cannonball momentum of v c m February 28, 2008 Physics for Scientists&Engineers 1 4 After firing the first cannonball, the mass of the rocket is reduced to m- m The rocket then receives a recoil momentum opposite to that of the cannonball The velocity change of the rocket is then And the total velocity of the rocket is Toy Model of a Rocket (2) Toy Model of a Rocket (2) p r = ! v c " m ! v 1 = " v c ! m m " ! m 1 1 1 v v v = + ! = ! February 28, 2008 Physics for Scientists&Engineers 1 5 Toy Model of a Rocket (3) Toy Model of a Rocket (3) Now we fire the second cannonball Firing the second cannonball reduces the mass of the rocket from m- m to m- 2 m and gives the rocket an addition recoil velocity of ! v 2 = " v c ! m m " 2 ! m February 28, 2008 Physics for Scientists&Engineers 1 6 Toy Model of a Rocket (4) Toy Model of a Rocket (4) Firing the n th cannonball gives the rocket a velocity change of So the velocity of the rocket after firing the n th cannonball is We recognize this expression as a recursion relation We can solve this type of equation exactly numerically, which can be fairly tough ! v n = " v c ! m m " n ! m v n = v n ! 1 + " v n February 28, 2008 Physics for Scientists&Engineers 1 7 Toy Model of a Rocket (4) Toy Model of a Rocket (4) Firing the n th cannonball gives the rocket a velocity change of So the velocity of the rocket after firing the n th cannonball is We recognize this expression as a recursion relation We can solve this type of equation exactly numerically To make our life easier , lets make a well-justified simplifying assumption ! v n = " v c ! m m " n ! m v n = v n ! 1 + " v n February 28, 2008...
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W8.4 - February 28, 2008 Physics for...

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