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

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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: v ( t ) v o e kv dv = - C m t 0 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) x ( t ) 0 dx = 1 k t 0 dt ln - Ck m t + e kv o (9) relabeling a = Ck m and b = e kv o , and note 1 k t 0 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

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To calculate the stopping distance, x s , note that t s = 1 a ( b - 1) in our notation: we get x s = 1 ka [ln( b ) b - b + 1 + 0] (12) or x s = m k 2 C [ln( e kv o ) e kv o - e kv o + 1] = m k 2 C [ e kv o ( kv o - 1) + 1] (13) 3. Integrating F = ma = m dv dt = - eE 0 sin( ωt ) (14) gives v ( t ) = eE 0 [cos( ωt ) - 1] (15)
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s2 - Phys 263/Amath 261 Assignment 2 SOLUTIONS 1 The law of...

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