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I read what if we have acceleration given as a function of velocity we can calculate time as

$$t(v) = t_0 + \int_{v_0}^{v} \frac{dv}{a(v)}.$$

Why?

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2 Answers 2

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You have $a(v) = \frac{dv}{dt}$. By separating the variables you get $dt= \frac{dv}{a(v)}$. Now you just integrate between $v_0$ and $v$ to obtain the equation you wrote.

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Really @Yola answered your question directly so I want to be a little more complete. I want to show how to deal with problems of $a(v)$ and $a(x)$.

  1. Acceleration as a function of speed $$ t= t_0 + \int_{v_0}^v \frac{1}{a(v)}\,{\rm d} v \\ x = x_0 + \int_{v_0}^v \frac{v}{a(v)}\,{\rm d}v $$
  2. Acceleration as function of position/displacement $$ \frac{v^2}{2} - \frac{v_0^2}{2} = \int_{x_0}^x a(x)\,{\rm d} x \\ t -t_0 = \int_{x_0}^x \frac{1}{v(x)} \,{\rm d}x $$
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