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

In summary, the question asks for the mass of a body before it disintegrated into two pieces with masses of 3.0kg and 4.0kg and velocities of 0.8c and 0.6c respectively. Using the energy-momentum relation and the relativistic kinetic and momentum equations, the attempt at a solution reveals that the mass-energy of the two pieces was not taken into account, resulting in an incorrect answer of 3kg. The correct answer, 10kg, is obtained by considering the mass-energy of the two pieces as well.
  • #1
Je m'appelle
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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
[tex]E^2 = (pc)^2 + (m_0 c^2)^2 [/tex]

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

3. Relativistic momentum equation
[tex]p_i = \frac{m_i v_i}{\sqrt{1-\frac{v_i^2}{c^2}}} [/tex]

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

[tex] 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) [/tex]

This yields [itex]m_0 = 3 \ kg[/itex] which doesn't make sense.

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

Any hints?
 
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  • #2
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.
 
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Likes Je m'appelle and PeroK
  • #3
tomdodd4598 said:
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
 

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

1. What is Relativistic Kinematics?

Relativistic Kinematics is a branch of physics that studies the motion and interactions of objects traveling at speeds close to the speed of light. It is based on the principles of special relativity, which describes how space and time are affected by the movement of objects.

2. How is Relativistic Kinematics different from Classical Kinematics?

Classical Kinematics, also known as Newtonian Kinematics, is based on the laws of motion proposed by Sir Isaac Newton. It is applicable to objects moving at speeds much slower than the speed of light. Relativistic Kinematics takes into account the effects of special relativity, such as time dilation and length contraction, which become significant at high speeds.

3. What is the Lorentz Transformation in Relativistic Kinematics?

The Lorentz Transformation is a mathematical equation that describes how quantities such as time, length, and velocity change for an observer when an object is moving at high speeds. It is an essential tool in calculating the effects of relativity in kinematic equations.

4. How does Relativistic Kinematics explain the Twin Paradox?

The Twin Paradox is a thought experiment in which one twin travels at high speed to a distant location while the other twin remains on Earth. When the traveling twin returns, they have aged less than the twin who stayed on Earth. Relativistic Kinematics explains this paradox by showing that time dilation occurs for the traveling twin due to their high speed relative to Earth.

5. What are some real-world applications of Relativistic Kinematics?

Relativistic Kinematics is used in many areas of modern physics, such as particle accelerators, nuclear reactions, and astrophysics. It also has practical applications in technologies such as GPS, where the effects of relativity must be taken into account for accurate measurements. Additionally, the principles of relativity and kinematics are essential in the development of theories like the Standard Model of particle physics and the theory of general relativity.

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