91Ó°ÊÓ

The Work-Energy Principle is a fundamental concept in physics that states that the work done on an object is equal to the change in its kinetic energy. This principle can be expressed mathematically as:

• Work Done (W) = Final Kinetic Energy (KE_f) - Initial Kinetic Energy (KE_i)

In simpler terms, when you apply a force to an object and it moves, you're doing work on the object, which results in a change in its kinetic energy. This principle applies universally, whether you're dealing with everyday speeds or high-speed particles like electrons.
In the realm of classical physics, kinetic energy calculations are straightforward. However, as velocities approach the speed of light, classical physics is no longer sufficient. Instead, we enter the domain of relativistic physics, where the computations and principles require adjustments to incorporate the effects predicted by Einstein's Theory of Relativity.

Relativistic Physics comes into play when dealing with velocities that are a significant fraction of the speed of light. Classical mechanics assumptions break down, and the principles of relativity must be considered. According to Einstein’s Theory of Relativity, mass increases with speed, leading to adjustments in kinetic energy calculations.
This means the relativistic kinetic energy can't be calculated using the simple \( KE = \frac{1}{2}mv^2 \) formula. Instead, it involves a correction factor accounting for the velocity's proximity to the speed of light (c), expressed as:

• \( KE = \left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right)m_ec^2 \)

where:
- \( m_e \) is the rest mass of the electron.
- \( v \) is the velocity of the electron.
These corrections ensure that calculations like those for an electron's change in speed adhere to the relativistic restrictions that keep velocities from surpassing the speed of light. This consideration is vital in high-speed environments, such as particle accelerators.

Kinetic energy calculation in a relativistic context requires understanding both how speed changes affect energy and how to apply the relativistic formula accurately. To calculate the work done to increase an electron's speed, follow these steps:

• Determine initial and final velocities in terms of the speed of light \( (v_i, v_f) \).
• Plug these velocities into the relativistic kinetic energy formula \( KE = \left(\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}-1\right)m_ec^2 \).
• Calculate the kinetic energy for both initial and final speeds.
• Subtract the initial kinetic energy from the final kinetic energy to find the work done (\( W = KE_f - KE_i \)).

In the exercise, the velocities were given as fractions of \( c \), specifically \( 0.18c \) to \( 0.19c \) and \( 0.98c \) to \( 0.99c \). Despite the same speed increment, the energy required to change the electron's speed increases substantially as it nears the speed of light because the relativistic effects become more pronounced.