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When we study the motion of objects at speeds much less than the speed of light, we use the principles of classical mechanics, which are part of a broader theme known as nonrelativistic physics. One of the fundamental concepts in this area is nonrelativistic momentum, which serves as a measure of an object's motion. We often define the momentum (represented by the symbol p) of a particle as the product of its mass m and its velocity u, expressed by the simple equation p = m u.

This equation suggests that momentum is directly proportional to both mass and velocity—a more massive or faster-moving object will carry more momentum. However, as velocity approaches a significant fraction of the speed of light, this simplistic definition fails to hold true. In educational contexts, it's essential to comprehend this straightforward concept first, to then grasp why and how it evolves when we move into the realm of relativistic physics.

In physics, the speed of light (c) is a critical universal constant and sets a limit on how fast information or matter can travel through space. The speed of light is approximately 299,792,458 meters per second. In the context of momentum, the speed of light is not just a speed cap but also a demarcation line between nonrelativistic (everyday speeds) and relativistic speeds (speeds close to c).

When objects move at a significant fraction of c, we observe deviations from the predictions of classical mechanics, like time dilation and length contraction. Understanding the speed of light is crucial both for theoretical insights and practical applications, such as plotting satellite orbits or synchronizing GPS systems.

The realm of relativistic physics, championed by Albert Einstein's theory of relativity, modifies our understanding of motion when dealing with objects traveling at speeds close to c. It introduces the concept of relativistic momentum, where the classical definition p = m u gets replaced by a more complex equation that takes into account the Lorentz factor—\frac{1}{\(1-\frac{u^2}{c^2}\)^{\frac{1}{2}}}. The relativistic momentum expression is p = \frac{m u}{\sqrt{1-\frac{u^2}{c^2}}}. This takes into account the effects of relativistic speeds on mass and time.

The key takeaway here is that the faster an object moves, the more pronounced the effects of relativity become, leading to an increase in momentum that's not linear with speed, unlike the nonrelativistic scenario. This topic is profound as it underpins the workings of particle accelerators and the behavior of cosmic particles.

Whenever we measure and calculate momentum at speeds approaching the speed of light, we must account for potential errors arising from using nonrelativistic formulas. Accurately calculating such errors is critical when precision is essential, like in experimental physics or engineering. For instance, the problem at hand requires finding at what velocities the use of the nonrelativistic equation gives an error in the momentum of 1.00% and 10.0%.

To perform this momentum error calculation, the relativistic momentum is compared to the nonrelativistic momentum, and the discrepancies are expressed as percentages. By setting up and solving equations with error percentages factored in, one can find the specific velocities at which these errors occur. This process helps foster a deeper understanding of the transition between classical and relativistic mechanics, and is fundamental for students as they explore realms wherein every fraction of a percent can lead to dramatically different outcomes.