In the theory of relativity, the mass of a particle with velocity v is
m = m0 / sqrt(1-v^2 / c^2)
where m0 is mass of the particle at rest and c is the speed of light. What happens as v approaches c^-?
mass is unbounded, it increases. Since this cannot happen, velocity cannot equal the speed of light.
As v approaches c (the speed of light) from below, meaning v gets closer and closer to the speed of light, let's analyze what happens to the equation:
m = m0 / sqrt(1 - v^2 / c^2)
As v approaches c^- (approaching the speed of light from below), the term v^2 / c^2 becomes very close to 1.
When v^2 / c^2 is close to 1, subtracting it from 1 gives a very small value. Taking the square root of this value will result in a very large number.
Therefore, as v approaches c^-, the denominator of the equation becomes extremely close to zero, resulting in the mass (m) approaching infinity. In other words, the mass of the particle increases without bound as the particle's velocity approaches the speed of light.
This phenomenon is known as relativistic mass increase, where the mass of an object appears to increase as it approaches the speed of light. It is a characteristic effect predicted by the theory of relativity.