# Derivation of Relativistic Kinetic Energy and Total Energy

In classical mechanics, the mass of a moving particle is independent of its velocity. But in special theory of relativity one can see that mass is also relative.

In the special theory of relativity, length, time, velocity and mass is relative. If these variables are relative the Kinetic energy and hence total energy will be relative. In this lecture I just try to define and explain that how to derive result for relativistic kinetic energy and total energy.

As the topic starts from a particle which is kept in rest and mass of that is ” m0 “, when a force is applied on it, the particle starts to move. The speed of the particle depends on the magnitude of the force, if force increases it speed will increase and hence its kinetic energy. It means the work done on the particle is basically its kinetic energy. More force, more work and more kinetic energy. By using this concept, one can derive the relation for particle’s kinetic energy and total energy.

In beginning, a particle is in rest and when a force act on it, particle starts to move, the mass of that particle is now ” m ” and speed is ” v “. The work done to displace this particle by distance “dx”, will be ” F.dx “. This work on the particle appear in terms of the small amount of kinetic energy. if suppose particle displace from position A to B wich is a distance ” x ” then the total amount of work done will be the integration of it from ” 0 ” to “x ” into the right hand side. This will be the total kinetic energy of the particle and now by simple differentiation and integration tricks one can solve the mathematics.

At the end we get KE= mc^2 + m0C^2; where mc^2 is the total energy which is represented by ” E” and second term is called as rest mass energy ( the potential energy) of the particle.

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