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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