Question

at what speed the mass of proton be doubled

Answer 1

The mass of a proton at speed v is

$$m=2m_0=\frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}}$$

$$2=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\to 1-\frac{v^2}{c^2}=0.25$$

$$v=\frac{\sqrt{3}}{2}c=2.598\cdot10^8\frac{m}{s}$$