What is the velocity of an electron that has a momentum of \(3.04 \times 10^{-21} \mathrm{kg} \cdot \mathrm{m} / \mathrm{s}\) ? Note that you must calculate the velocity to at least four digits to see the difference from \(c\)

To understand how to calculate the velocity of an electron when given its momentum, we should first address what each term represents. Momentum, denoted by the symbol 'p', is a measure of the quantity of motion an object has and can be found by multiplying the mass 'm' of an object by its velocity 'v'. In our exercise, we're given the electron's momentum and we know the mass of an electron to be approximately equal to 9.11 x 10^-31 kilograms.

With these knowns, we proceed to calculate the velocity of an electron using the following steps:

• Start by writing down the momentum-velocity formula: p = mv.
• Rearrange the formula to isolate 'v': v = p/m.
• Substitute the given momentum and the known mass of the electron into the rearranged formula.

Following these steps provides a clear pathway to finding the electron's velocity, demonstrating the usefulness of algebraic manipulation in physics problems.

The momentum-velocity relationship is fundamental in mechanics. According to the principle of conservation of momentum, the total momentum of a closed system is constant in the absence of external forces. In the case of an electron, its momentum is productively related to its velocity, which means if we know the momentum and the mass of the electron, we can easily find its velocity.

Momentum is a vector quantity, which means it has both magnitude and direction, but for the purpose of the exercise provided, we are concerned with the magnitude. By rearranging the momentum equation, \( p = mv \) to \( v = p/m \) we effectively use this relationship to calculate one variable when the other two are known. This allows us to understand how fast an electron is moving (its velocity) given information about how much motion it possesses (its momentum).

In many calculations in physics, physical constants play a crucial role. These constants are measurable properties of the natural world that remain constant in all cases, such as the gravitational constant, speed of light, and the electron mass. When calculating an electron's velocity from its momentum, the constant we reference is the mass of the electron, which is approximately \(9.11 \times 10^{-31}\) kg.

The precision of these constants is important as it ensures the accuracy of our calculations. In our given problem, the exercise advised that the velocity be calculated to at least four digits to appreciate the difference from the speed of light 'c'. This highlights the relationship between precise constants and the specifics of a calculation that could change significantly with even the slightest variance in the values used. Such care is crucial in scientific computations where exactness is both invaluable and indispensable.