Falling object and the air resistance

Case(1):

$m \dfrac{dv}{dt} = mg - k_{1}v \implies \dfrac{dv}{dt} = g - av,$ where $a = \dfrac{k_{1}}{m} \implies$ $$

$\dfrac{dv}{g - av} = dt \implies -\dfrac{1}{a} \ln(g - av) = t+ c \implies$ $$

$g - av = Ce^{-at} \implies v(t) = \dfrac{1}{a}(g - Ce^{-at})$ $$

$v(0) = 0 \implies C = g \implies v(t) = \dfrac{g}{a}(1 - e^{-at})$ $$

The terminal velocity $V_{1} = lim_{t \rightarrow \infty} v(t) = \dfrac{g}{a} = \dfrac{mg}{k_{1}}$ $$

Case(2): $$

$m \dfrac{dv}{dt} = mg - k_{2}v^2 \implies \dfrac{dv}{dt} = g - a^2v,$ where $a^2 =\dfrac{k_{2}}{m} \implies$

$\int \dfrac{dv}{g - a^2v^2} = \int dt + c$ $$

Solving $\int \dfrac{dv}{g - a^2v^2}$ using partial fractions. $$

$\dfrac{1}{(\sqrt{g} - av)(\sqrt{g} + av)} = \dfrac{A}{\sqrt{g} - av} + \dfrac{B}{\sqrt{g} + av} \implies$ $$

$1 = (A + B) \sqrt{g} + a * (A - B) v \implies$

$A + B = \dfrac{1}{\sqrt{g}}$ $A - B = 0$

$\implies A = \dfrac{1}{2 \sqrt{g}} = B \implies \dfrac{1}{2 \sqrt{g}} \int \dfrac{1}{\sqrt{g} - av} + \dfrac{1}{\sqrt{g} + av} \: dv = \dfrac{1}{2 \sqrt{g}} \ln(\dfrac{\sqrt{g} + av}{\sqrt{g} - av}) = t + c \implies$ $$ $\sqrt{g} + av = C e^{2a\sqrt{g} t} (\sqrt{g} - av) \implies$ $$

$v(t) = \dfrac{\sqrt{g}}{a} (\dfrac{C e^{2a\sqrt{g} t} - 1}{C e^{2a\sqrt{g} t} + 1})$ $$

$v(0) = 0 \implies C = 1 \implies v(t) = \dfrac{\sqrt{g}}{a} (\dfrac{e^{2a\sqrt{g} t} - 1}{e^{2a\sqrt{g} t} + 1})$ $$

or $v(t) = \dfrac{\sqrt{g}}{a} (\dfrac{1 - e^{-2a\sqrt{g} t}}{1 - e^{-2a\sqrt{g} t}})$ $$

The terminal velocity $V_{2} = lim_{t \rightarrow \infty} v(t) = \dfrac{\sqrt{g}}{a} = \sqrt{\dfrac{mg}{k_{2}}}$ $$

$\dfrac{V_{2}}{V_{1}} = 1 \implies \dfrac{k_{1}}{\sqrt{k_{2}}} = \sqrt{mg}$ $$

Using $m = 10$ and $g = 10 \implies \dfrac{k_{1}}{\sqrt{k_{2}}} = \boxed{10}.$

When body has reached terminal velocity, $a=\frac{dv}{dt}=0$ So by Newton's Law, $mg-k_1v_1=m*0$ So $k_1=\frac{mg}{v_1}$

In case 2, $mg-k_2 {v_2}^2=0$ so $k_2=\frac{mg}{{v_2}^2}$

$v_1=v_2$ So $\dfrac{k_{1}}{\sqrt{k_{2}}}=\frac{mg}{\sqrt{mg}}=\sqrt{mg}=\sqrt{100}=10$

Thus our answer is 10.