高中物理(二上)
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Chap 2 本章難題練習解答

難題 1. \frac{\sqrt{5}\; \ell}{v}

類題 1. \frac{\sqrt{3}\; \ell}{v}

類題 2. \ell\sqrt{\ell \,^2\; +\; 4\, \pi \,^2\, r\,^2}

類題 3. \sqrt{\frac{\ell \,^2}{4}\; +\; 4\, r\, ^2}\frac{1}{2}\, \sqrt{\ell \,^2\; +\; 4\, \pi \,^2\, r\,^2}

類題 4. \sqrt{\frac{\ell \,^2}{16}\; +\; 2\, r\, ^2}\frac{1}{4}\, \sqrt{\ell \,^2\; +\; 4\, \pi \,^2\, r\,^2}

難題 2. v\, \sec\, \theta

類題 1. v\; =\; u\, \cos\, \theta

【解法1】v\, t\; =\; s\, _1\; -\; s\, _2\; =\; \frac{h}{\sin \, \theta \, _1}\; -\; \frac{h}{\sin \,\theta \, _2}\; =\; \frac{h\, (\sin \, \theta \, _2\; -\; \sin \, \theta \, _1)}{\sin \, \theta \, _1\, \sin \, \theta \, _2}\;\; \cdots \cdots \; \; (1)
      u\, t\; =\; \frac{h}{\tan \, \theta \, _1}\; -\; \frac{h}{\tan \, \theta \, _2}\; =\; \frac{h\, (\tan \, \theta \, _2\; -\; \tan \, \theta \, _1)}{\tan \, \theta \, _1\, \tan \, \theta \, _2}\; =\; \frac{h\, (\frac{\sin \, \theta \, _2}{\cos \, \theta \, _2}\; -\; \frac{\sin \, \theta \, _1}{\cos \, \theta \, _1})}{\frac{\sin \, \theta \, _1}{\cos \, \theta \, _1}\; \times \; \frac{\sin \, \theta \, _2}{\cos \, \theta \, _2}}
         =\; \frac{h\, (\sin \, \theta \, _2\, \cos \, \theta \, _1\; -\; \cos \, \theta \, _2\, \sin \, \theta \, _1)}{\sin \, \theta \, _1\, \sin \, \theta \, _2}\; =\; \frac{h\, \sin \, (\theta \, _2\; -\; \theta \, _1 )}{\sin \, \theta \, _1\, \sin \, \theta \, _2}\;\; \cdots \cdots \; \; (2)
    \frac{(1)}{(2)}\;\; \Rightarrow \quad \frac{v\, t}{u\, t}\; =\; \frac{v}{u}\; =\; \frac{\sin \, \theta \, _2\; -\; \sin \, \theta \, _1}{\sin \, (\theta \, _2\; -\; \theta \, _1 )}\; =\; \frac{\sin \, (\theta \; +\; \Delta \theta)\; -\; \sin \, \theta}{\sin \, \Delta \theta}
      =\; \frac{\sin \, \theta\, \cos \, \Delta \theta\; +\; \cos \, \theta\, \sin \, \Delta \theta\; -\; \sin \, \theta}{\sin \, \Delta \theta}\; =\; \cos \, \theta\; +\; \sin \, \theta\, \frac{\cos \, \Delta \theta \; -\; 1}{\sin \, \Delta \theta}
      =\; \cos \, \theta\; +\; \sin \, \theta\, \frac{(1\; -\; 2\, \sin \, ^2\, \frac{\Delta \theta}{2})\; -\; 1}{2\, \sin \, \frac{\Delta \theta}{2}\, \cos \, \frac{\Delta \theta}{2}}\; =\; \cos \, \theta\; -\; \sin \, \theta\, \tan \, \frac{\Delta \theta}{2}
    當 \Delta \theta \; \rightarrow \; 0\quad \Rightarrow \quad \tan \, \frac{\Delta \theta}{2}\; \rightarrow \; 0 ,代入上式可得 \frac{v}{u}\; \doteq \cos \, \theta
    ∴ v\; =\; u\, \cos \, \theta

 

0最後修改紀錄: 2009/10/08(Thu) 14:34:02


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since 2011/06/20 18:23