Q77 / 120 · VITEEE 2018 · Mathematics

The solution of $\frac{d v}{d t}+\frac{k}{m} v=-g$ is

  • Correct answer
    A
    $v=c e^{-\frac{k}{m} t}-\frac{m g}{k}$
  • B
    $v=\mathrm{c}-\frac{m g}{k} e^{-\frac{k}{m} t}$
  • C
    $v e^{-\frac{k}{m} t}=c-\frac{m g}{k}$
  • D
    $v e^{\frac{k}{m} t}=\mathrm{c}-\frac{m g}{k}$

Solution

$$ \begin{array}{l} \frac{d v}{d t}+\frac{k}{m} v=-g \Rightarrow \frac{d v}{d t}=-\frac{k}{m}\left(v+\frac{m g}{k}\right) \\ \Rightarrow \frac{d v}{v+m g / k}=-\frac{k}{m} d t \Rightarrow \log \left(v+\frac{m g}{k}\right) \\ =-\frac{k}{m} t+\log C \\ \Rightarrow \mathrm{v}+\frac{m g}{k}=C e^{-k t / m} \Rightarrow v=C e^{-k t / m}-\frac{m g}{k} \end{array} $$

Hence, the correct answer is option: 'A'.

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