Question

By what factor does the average velocity of a gaseous molecule increase when the temperature in kelvin id doubled

Answer 1

The average velocity of a gaseous molecule increase when the temperature in kelvin is doubled becomes 1.4 kelvin.

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Answer 2

we know,
average velocity of gaseous molecule is given by $\overline{v}=\sqrt{\frac{8RT}{\pi M}}$

here you can see that average velocity is directly proportional to square root of temperature in Kelvin.
e.g., $v\propto \sqrt{T}$

now, temperature is doubled , then average velocity of gaseous molecule is $\overline{v'}\propto\sqrt{2T}$

change in average velocity, $\overline{v'}-\overline{v}=\Delta\overline{v}=K(\sqrt{2}-1)\sqrt{T}$
here, K is proportionality constant.

so, $\Delta\overline{v}=K(0.414)\sqrt{T}$

now, % percentage change in average velocity
$= \frac{K(0.414)\sqrt{T}}{K\sqrt{T}}\times100$
= 41.4 %