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Momentum is a key concept in both classical and relativistic mechanics. It provides a measure of how much motion an object has and is the product of the object's mass and velocity. In classical mechanics, momentum is given as \( p = mv \), where \( p \) represents momentum, \( m \) is the mass, and \( v \) is the velocity.
This formula works well when dealing with speeds much lower than the speed of light. However, at higher velocities, especially those approaching the speed of light, relativistic effects become significant.
In such cases, momentum is calculated using relativistic mechanics to ensure accuracy.

Nonrelativistic calculations involve using classical physics to describe motion. Specifically, these calculations ignore relativistic effects which are insignificant at low speeds compared to the speed of light. In the exercise, the nonrelativistic momentum is calculated using \( p = mv \), assuming velocities that don't approach the speed of light.
This simplification works well for terrestrial and many everyday phenomena. However, it breaks down for particles moving at significant fractions of the speed of light, such as those in particle accelerators.
Recognizing when nonrelativistic calculations are applicable is essential for accurate computations systems like engineering and mechanics.

The speed of light, denoted by \( c \), is approximately \( 3 \times 10^8 \) meters per second. This fundamental constant is crucial in the study of relativistic mechanics. It serves as the maximum speed limit in the universe and affects how physical processes are perceived at high velocities.
When objects move at speeds close to \( c \), their behavior diverges significantly from classical predictions. This is due to time dilation, length contraction, and the relativity of simultaneity.
Understanding the role of the speed of light is vital when approaching problems that require relativistic calculations, as in the given exercise.

The difference between relativistic and nonrelativistic momentum arises from the effects of relativity on high-speed motion. While nonrelativistic momentum is straightforward with \( p = mv \), this approximation fails at speeds nearing the speed of light. Relativistic momentum takes the form \( p = \frac{mv}{\sqrt{1-\frac{v^2}{c^2}}} \).
This equation incorporates a correction factor \( \sqrt{1-\frac{v^2}{c^2}} \), which becomes significant as velocity \( v \) approaches \( c \). In the exercise, we're asked at what velocity the momentum calculated relativistically differs by 1% from the nonrelativistic approach, illustrating the validity of both methods.
Such explorations help us understand the precise nature of motion at different scales, bridging classical and modern physics.