The temperature of an ideal gas is increased from 200K to 800K . If...

At $200K$, the rms speed is given by

${v}_{0} = \sqrt{\dfrac{3R \times 200}{M}} . . . ( 1 )$

At $800K$, the rms speed is given by

$v ' = \sqrt{\dfrac{3R \times 800}{M}} . . . \left(2\right)$

Divide equation (2) by equation (1) and solve to calculate the required speed.

$\begin{cases}\dfrac{v '}{{v}_{0}} & = & \dfrac{\sqrt{\dfrac{3R \times 800}{M}}}{\sqrt{\dfrac{3R \times 200}{M}}} \\ & = & \sqrt{\dfrac{800}{200}} \\ & = & 2 \\ & ⇒ & v ' = 2{v}_{0}\end{cases}$