Please show me how to derive the relativist mass formula:
m = m0 / sqrt(1 - v^2/c^2)?
It is not a trivial derivation.
http://www.softcom.net/users/greebo/lorentz.htm
To derive the relativistic mass formula, we will use the principles of special relativity. Let's start with the equation for the total energy of an object, which includes both its rest energy and its kinetic energy:
E = mc^2
Here, E represents the total energy, m is the mass, and c is the speed of light in a vacuum.
Now, let's consider an object that is moving with a velocity v relative to an observer. The observer sees the object having a kinetic energy given by:
KE = (1/2)mv^2
To account for the relativistic effects, we need to consider that the total energy E is related to the rest energy E0 (the energy of an object at rest, which is equivalent to its rest mass energy) and the kinetic energy KE. According to special relativity, the total energy E is related to the rest energy E0 by the equation:
E^2 = E0^2 + (m0c^2)^2
Here, m0 represents the rest mass of the object.
By substituting the expressions for E and E0, we get:
(mc^2)^2 = (m0c^2)^2 + (1/2)mv^2
Expanding and rearranging the equation, we find:
m^2c^4 = m0^2c^4 + (1/2)mv^2c^2
Dividing both sides by c^4, we obtain:
m^2 = m0^2 + (1/2)mv^2/c^2
Now, let's simplify this equation. Assuming m ≠ 0, we can rearrange the terms as follows:
m^2 = m0^2(1 + (1/2)v^2/c^2)
Dividing both sides by m0^2, we get:
(m^2/m0^2) = 1 + (1/2)v^2/c^2
Taking the square root of both sides, we find:
m/m0 = sqrt(1 + (1/2)v^2/c^2)
Finally, rearrange this equation to get the relativistic mass formula:
m = m0 / sqrt(1 - v^2/c^2)
Therefore, the relativistic mass formula is derived from the principles of special relativity and the consideration of total energy, rest energy, and kinetic energy.