# How fast?

Given that:

$E^2 = \left(\frac{m_0c^2}{\sqrt{1 - \frac{v^2}{c^2}}}\right)^2 + \left(\frac{m_0vc}{\sqrt{1 - \frac{v^2}{c^2}}}\right)^2$

How fast would an object with a rest mass of $$1 \text{ Pg}$$ need to be going to have the same 'mass' as an object with a rest mass of $$5.5 \text{ Pg}$$?

Give your answer as a fraction of $$c$$ to $$6\text{.d.p}$$. For example, if you got $$37,011,178 \text{ ms}^{-1}$$ then you would answer with $$0.123456$$ as $$37,011,178 \text{ ms}^{-1} = 0.123456 c$$ to $$6\text{.d.p}$$.

Details and Assumptions

• $$E$$ is energy in $$J$$.
• $$m_0$$ is rest mass in $$\text {Kg}$$.
• $$v$$ is velocity in $$\text{ms}^{-1}$$.
• 'Mass' refers to an objects mass-energy.
• Assume the second object has only mass-energy and no other types.
• $$c = 299,792,458 \text{ ms}^{-1}$$.
• $$1 \text{ Pg} = 10^{12} \text{ Kg}$$.
• The equation at the top of the page is the complete mass-energy equivalence equation.
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