s2 - Phys 263/Amath 261 Assignment 2 SOLUTIONS 1. The law...

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Unformatted text preview: Phys 263/Amath 261 Assignment 2 SOLUTIONS 1. The law of conservation of mass does not hold in quantum mechanics and special relativity. 2. The Force differential equation is defined by F = ma = m dv dt =- Ce- kv (1) in one dimension. Integrating: Z v ( t ) v o e kv dv =- C m Z t dt (2) 1 k e kv v ( t ) v o =- C m t (3) e kv ( t )- e kv o =- Ck m t (4) v ( t ) = 1 k ln- Ck m t + e kv o (5) It is easy to solve for the time the boat takes to come to a stop t s , defined by the condition v ( t s ) = 0, from Eq. 4: 1- e kv o =- Ck m t s (6) t s = m Ck ( e kv o- 1 ) (7) To get the position x ( t ), integrate the velocity solution: dx dt = 1 k ln- Ck m t + e kv o (8) Z x ( t ) dx = 1 k Z t dt ln- Ck m t + e kv o (9) relabeling a = Ck m and b = e kv o , and note 1 k Z t dt ln (- at + b ) = 1 a [ln( b ) b + ln(- at + b ) ta- ln(- at + b ) b- at ] (10) we get x ( t ) = 1 ka [ln( b ) b- at + ln(- at + b ) ( ta- b )] (11) 1 To calculate the stopping distance, x s , note that t s = 1 a ( b...
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This note was uploaded on 09/20/2010 for the course AMATH 261 taught by Professor Rogermelko during the Spring '10 term at Waterloo.

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s2 - Phys 263/Amath 261 Assignment 2 SOLUTIONS 1. The law...

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