Question

Why Newtonian mechanics doesn't work at velocities close to that of the speed of light? I know that mass tends to become infinite at v=c but why does it fail if v is close to c but not exactly c?

Answer 1

At velocities that are much higher than what we expect in our everyday life, the classical laws given by Newton are replaced by the laws of special relativity given by Albert Einstein. As an object moves at high speeds, its kinetic energy is added as mass and thus the mass of the moving body tends to increase with increasing velocity. The relation of mass and velocity is given as,

where,

• m(v) is the mass of object in motion
• is the rest mass of object
• v is the velocity of object
• c is the speed of light

In relativistic mechanics, momentum is defined as,

Here we may see that in classical mechanics, momentum rises linearly with increasing ratio of v/c while in relativistic mechanics, the momentum will rise steeply with the increasing value of v/c. Thus even when v is close to c (but not exactly c) there will be significant difference between the classically defined momentum and the relativistically defined momentum. Any mass that is gained with rise in velocity will result in change in the momentum of the body even when there is no application of force. This result contradicts Newton's laws of motion and leads to the failure of classical laws at high speeds.

We may also study the Relativistic second law to understand this difference better. In relativity, Newton's second law is given as,

F = dP/dt

using the relation for momentum in relativity,

The acceleration (a) of objects given as,

Even when the force is constant, the acceleration of particle decreases as its velocity increases and as This is contrary to the classical laws which predict that if there is no application of force on a body, the acceleration of object cannot change. However, at high velocities, we expect the acceleration of objects to decrease with increasing velocities as no object can move with speed of light. This prediction is in accordance with the Special Theory of Relativity.

Thus it turns out that Newton's laws are just an approximation and do not work at high speeds due to relativistic effects.