1746-IM4（A-B）
1746-IM4 我們要意識到一件事：前面我們都假設力F是恒力，認為物體在做勻加速運動，這是一種特例。
我們要計算物體的動能，要推導質能方程，當然不希望它只在這種特殊情況下才成立。所以，我們要考慮更一般的情況：如果力F和位移S都在變，我們應該如何計算它們的乘積？
地球的表面是彎的，但在小范圍內我們可以認為它是平的。同理，在足夠小的范圍內，我一樣可以認為力F和位移S的大小不變。如果用ds表示這個微小的位移變化，用F·ds表示力F在這個微小位移里做的功，那么，把0到S所有的功累加起來就能得到總動能E。
寫成數學表達式就是這樣：
很顯然，為了保證結果的一般性，我們這里動用了微積分。這個具體的計算過程我不想多講，因為但凡學了微積分，會分部積分的同學都知道怎么算。如果你不會微積分，這個計算過程我也沒法在這里給你科普，我只能建議你先看看我的《你也能懂的微積分》，再找本微積分教材看看。
更為重要的是：這個計算過程并不會影響你對質能方程的理解。
因為這只是一個純數學計算手段。人們之所以誤解質能方程，并不是因為不知道這個公式的形式是E=mc?，而是無法理解這個方程背后的物理意義和物理背景。
如果你跟著我的思路來到這里，知道為了讓動量守恒定律滿足洛倫茲協變性，我們不得不重新定義了動量，進而需要重新定義動能。你就會知道質能方程到底是怎么來的，就算看不懂中間的計算過程，也不會影響你對質能方程的理解。
這里，我就放一張新動能的推導圖片，你能看懂就看，看不懂也沒事。當然，如果你暫時看不懂，但是為了能看懂而去學習微積分，那自然是極好的。這里也沒多少微積分的知識，關鍵就是一個分部積分。計算思路也非常簡單，就是用狹義相對論里新動量的變化率代替力F：
我把結果放到倒數第二步：
也就是說，一個物體的動能E在狹義相對論里可以表示成這樣：括號外面是mc?，括號里面是相對論因子減去1。
我們把中間那一大串東西稱為相對論因子（也叫洛倫茲因子），因為相對論里經常會用到它，所以我們就用一個特殊符號γ來表示這個相對論因子：
這樣，你再看看狹義相對論里的新動量，是不是就相當于在牛頓力學的動量mv上乘了一個相對論因子γ？也就是說，狹義相對論里的新動量可以簡寫成p=γmv。
1746-IM4
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