A particle is moving at \(0.90 c .\) If its speed increases by \(10 \%,\) by what factor does its momentum increase?

Special relativity is a theory put forth by Albert Einstein that revolutionized the way we understand the relationship between space, time, and motion. One of its core claims is that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum is the same no matter the speed at which an observer travels.

Another intriguing aspect of special relativity is the idea of time dilation and length contraction. As an object moves closer to the speed of light, time will actually pass slower for it relative to a stationary observer, and its length will appear contracted in the direction of motion. These effects become more pronounced as you approach the speed of light, leading to some very counterintuitive phenomena.

The momentum increase factor is a concept derived from the theory of special relativity, reflecting how the momentum of an object changes as its velocity approaches the speed of light (c). While classical momentum is simply the product of mass and velocity, relativistic momentum takes into account the effects predicted by Einstein's theory.

In relativistic terms, momentum is no longer a linear function of velocity. Instead, it increases exponentially as the velocity of an object approaches the speed of light due to the factor \(\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}\), which arises from time dilation and length contraction. This means even a small increase in speed near the speed of light can result in a large increase in momentum.

The speed of light is commonly denoted by the symbol c, and it is a fundamental constant of nature. In a vacuum, light always travels at approximately 299,792,458 meters per second, irrespective of the observer's relative motion. This constancy is the bedrock upon which special relativity is built.

Due to cosmic speed limit set by the speed of light, as an object's speed increases and gets closer to c, effects like time dilation and mass energy equivalence emerge. These effects significantly impact the object's momentum as well. It's worth noting that nothing with mass can reach the speed of light because, as per Einstein's equations, it would require an infinite amount of energy to do so. The speed of light is thus not only a speed limit but also a key part of the framework governing the universe's structure and workings.