How Do You Calculate the Original Mass in a Relativistic Disintegration Problem?

[Je m'appelle]

Homework Statement

A body at rest in a frame of reference S disintegrates into two pieces moving in opposite directions. The masses of each fragment are 3.0kg and 4.0kg and their velocities 0.8c and 0.6c, respectively. Find the mass of the body before it disintegrated. (Answer: 10kg)

Homework Equations

1. Energy-momentum relation
$$E^2 = (pc)^2 + (m_0 c^2)^2$$

2. Relativistic kinetic energy equation
$$E_{ki} = m_i c^2 \left(\frac{1}{\sqrt{1-\frac{v_i^2}{c^2}}} -1 \right)$$

3. Relativistic momentum equation
$$p_i = \frac{m_i v_i}{\sqrt{1-\frac{v_i^2}{c^2}}}$$

The Attempt at a Solution

First I tried using conservation of energy, by taking the energy of the body at rest (1.), with p=0, and equating it to the sum of the kinetic energies of the two fragments (2.), which looked like this

$$m_0c^2 = m_1c^2 \left( \frac{1}{\sqrt{1 - \frac{v_1^2}{c^2}}} - 1 \right) + m_2c^2 \left( \frac{1}{\sqrt{1 - \frac{v_2^2}{c^2}}} - 1 \right)$$

This yields $m_0 = 3 \ kg$ which doesn't make sense.

Then it occurred to me that since I already have the answer $(10 kg)$, which was provided in the problem, and both fragments sum up to $7 kg$ then there's $3 kg$ of mass missing, so I'm clearly skipping something here.

Any hints?

 

[tomdodd4598]

As far as I can see, you seem to be ignoring the mass-energy of the two pieces that fly apart - I think you're only considering their kinetic energy.

 

[Je m'appelle]

As far as I can see, you seem to be ignoring the mass-energy of the two pieces that fly apart - I think you're only considering their kinetic energy.

You're absolutely right! Thank you, tomdodd4598