Question

An electron is moving at a speed such that it could circumnavigate the Earth at the equator in \(1 \mathrm{~s}\). (a) What is its speed, in terms of the speed of light? (b) What is its kinetic energy \(K ?(c)\) What percent error do you make if you use the classical formula to calculate \(K\) ?

Short answer

The speed of the electron in terms of the speed of light is \(0.1336 c\). The kinetic energy of the electron can be calculated using the relativistic and classical kinetic energy formulas, and the percent error between them will be the final part of the answer.

Step-by-step solution

Calculate the speed of the electron

The electron circumnavigates the earth in 1 second. The circumference of the earth at the equator can be approximated as \(40,075\) kilometers. Therefore, the speed of the electron can be calculated using the formula for speed which is \[Speed = \frac{Distance}{Time}\]Using this formula the speed of the electron is \(40,075 km/s\).

Express the speed in terms of the speed of light

The speed of light is approximately \(299,792 km/s\). Calling the speed of light \(c\), and the speed of the electron \(v\), we can express \(v\) in terms of \(c\) by the formula: \[v = \frac{v}{c}\]Therefore, the speed of the electron in terms of the speed of light is \(0.1336 c\).

Calculate the kinetic energy using the relativistic formula

The relativistic kinetic energy \(K\) can be calculated using the formula: \[K = mc^2(\frac{1}{\sqrt{1 - v^2/c^2}} - 1)\]where \(m\) is the mass of the electron, \(c\) is the speed of light, and \(v\) is the speed of the electron. Using this formula and substituting the values, the kinetic energy of the electron can be calculated.

Calculate the kinetic energy using the classical formula and find the percent error

The classical kinetic energy formula is: \[K_{classical} = \frac{1}{2}mc^2\]where \(m\) is the mass of the electron and \(v\) is the speed of the electron. After calculating this, the percent error can be determined by: \[Percent Error = \frac{|K_{relativistic} - K_{classical}|}{K_{relativistic}} \times 100%\]This will give the percent error if you use the classical formula to calculate the kinetic energy of the electron.

Key concepts

Speed of Light

Understanding the speed of light is essential for exploring the realm of relativistic physics. The speed of light, symbolized as 'c', is a fundamental constant of nature and is roughly equal to 299,792 kilometers per second (km/s). It represents the maximum speed at which all energy, matter, and information in the universe can travel. It is also the speed at which light propagates through a vacuum.

In our exercise, the speed of an electron is compared to this universal speed limit to highlight the relativistic effects that come into play at high velocities. When objects move at speeds comparable to the speed of light, classical mechanics fails to accurately describe their behavior, and instead, Einstein's theory of relativity must be used. Relativity introduces corrections for high-speed objects, resulting in different predictions for quantities like time, length, mass, and kinetic energy.

Classical Kinetic Energy Formula

The classical kinetic energy formula is an expression used in Newtonian mechanics to calculate the energy of motion of an object. It is given by:
\[K_{classical} = \frac{1}{2}mv^2\] where 'm' represents the mass of the object and 'v' its velocity. This formula assumes that the speeds involved are much slower than the speed of light and that the laws of physics are the same in all inertial frames.

However, when an object's speed approaches a significant fraction of the speed of light, the classical formula becomes less accurate, and relativistic effects must be considered. The exercise mentions the classical formula's inaccuracy at high velocities, prompting learners to use the relativistic formula to find the kinetic energy and then calculate the error introduced by using the classical approach.

Percent Error Calculation

Percent error is a commonly used statistical measure to determine the accuracy of a value in comparison to a true or accepted value. The formula to calculate percent error is:
\[Percent Error = \frac{|Accepted Value - Experimental Value|}{Accepted Value} \times 100%\]
In the context of our exercise, the 'accepted value' is the kinetic energy calculated using the relativistic formula, which is necessary since the electron is moving at a speed comparable to the speed of light. The 'experimental value' is the kinetic energy obtained using the classical formula. By substituting these two values into the percent error formula, students can quantify the inaccuracy, or error, incurred when using classical mechanics instead of relativistic mechanics for fast-moving particles like electrons.

Percent error calculation is a vital tool when comparing theoretical predictions with experimental observations or when evaluating the precision of various scientific methods.