Question

At what speed v will the mass of an object be 10% greater than its rest mass?

Answer 1

\[ \small{ \text{"It is not good to introduce the concept of the mass} \ M=m/ \sqrt{ 1- v^2/c^2}
\\ \text{of a moving body for which no clear definition can be given. It is better to}
\\ \text{introduce no other mass concept than the 'rest mass' } m \text{. Instead of introduc-}
\\ \text{ing } M \text{ , it is better to mention the expression for the momentum and energy}
\\ \text{ of a body in motion." }
\\ \hspace{50px} \text{-Einstein } } \]

Given the total energy of a an object in motion
\[E=\gamma mc^2 \quad \quad ; \quad \quad \gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\]
where its rest energy is at v=0
\[E_0 = mc^2\]
we can see that total energy goes directly with mass; therefore, the question can be reframed as "At what speed v will the total energy of the object be 10% greater than the rest energy?"

\[E=1.1E_0\]
\[\gamma mc^2 = 1.1 (mc^2)\]
\[\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=1.1\]
Solve for v

Answer 2

In France, we stick to Einstein's view, we have no other mass concept than 'rest mass' in our textbooks.