Question

What is the value of \(\gamma\) for a particle moving at a speed of \(0.800 c ?\)

Short answer

Answer: The value of \(\gamma\) for a particle moving at a speed of \(0.800c\) is 1.67.

Step-by-step solution

We need to use the Lorentz factor formula, or the relativistic factor formula, which is given by: \(\gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}}\) #Step 2: Plug in the given values#

The speed \(v\) is given as \(0.800c\). Substitute this value into the Lorentz factor formula: \(\gamma = \frac{1}{\sqrt{1-(\frac{0.800c}{c})^2}}\) #Step 3: Simplify the equation#

Notice that the speed of light \(c\) is in both the numerator and the denominator, which will cancel each other out: \(\gamma = \frac{1}{\sqrt{1-(0.800)^2}}\) #Step 4: Calculate the value of \(\gamma\)#

Square the speed of the particle inside the parentheses and then subtract it from 1: \(\gamma = \frac{1}{\sqrt{1-0.64}}\) #Step 5: Complete the calculation#

Simplify the expression inside the square root and then take the reciprocal: \(\gamma = \frac{1}{\sqrt{0.36}} = \frac{1}{0.6}\) #Step 6: Find the final value of \(\gamma\)#

Divide 1 by 0.6 to find the value of the relativistic factor: \(\gamma = 1.67\) The value of \(\gamma\) for a particle moving at a speed of \(0.800c\) is 1.67.

Key concepts

Lorentz Factor

Understanding the Lorentz factor is essential when studying high-speed physics. It's a number which becomes significant as an object approaches the speed of light. The Lorentz factor, denoted as \(\gamma\), is calculated using the formula \(\gamma = \frac{1}{\sqrt{1-(\frac{v}{c})^2}}\), where \(v\) represents the velocity of the object and \(c\) is the speed of light in a vacuum.

This factor accounts for the effects of time dilation, length contraction, and mass increase that happen as an object moves closer to the speed of light, according to special relativity. As the speed \(v\) gets closer to the speed of light \(c\), the Lorentz factor \(\gamma\) increases. When \(v\) is much less than \(c\), the Lorentz factor is approximately equal to 1, meaning relativistic effects are negligible. However, as \(v\) approaches \(c\), \(\gamma\) increases dramatically, signifying a substantial relativistic effect.

Special Relativity

Special relativity is a fundamental theory proposed by Albert Einstein that revolutionized the way we understand space and time. One of the core principles is that the laws of physics are the same for all non-accelerating observers, and that the speed of light in a vacuum is the same no matter the speed at which an observer travels.

According to this theory, as objects move faster and closer to the speed of light, time appears to slow down (time dilation), lengths appear to contract (length contraction), and mass appears to increase (relativistic mass). These effects are all interconnected and are grasped by implementing the Lorentz factor in calculations, to modify the classical equations of motion for relativistic speeds. Special relativity has been confirmed by numerous experiments and is a cornerstone of modern physics.

Speed of Light

The speed of light is known to be the universal speed limit. Denoted as \(c\), its value in a vacuum is exactly 299,792,458 meters per second. A fascinating aspect of this constant is that it remains the same regardless of the observer's motion relative to the light source, as stated in Einstein's theory of special relativity.

This invariant speed plays a crucial role in physics, tying together space and time into what Einstein called space-time. It also serves as a key component in the calculations of the Lorentz factor. The constancy of the speed of light leads to intriguing consequences like the fact that no matter how fast you are moving, light will always pass you at the same speed.

Relativistic Physics

Relativistic physics explores the behavior of objects moving at speeds comparable to the speed of light. This realm is governed by the principles of special relativity. Standard Newtonian mechanics falls short when dealing with such speeds, as classic equations do not factor in the relativistic effects and thus provide incorrect predictions.

To accurately describe high-speed phenomena, physicists must use modified formulas that incorporate the Lorentz factor. These adjustments allow accurate predictions of observed phenomena like time dilation in GPS satellites or particle accelerator experiments. Relativistic physics not only provides insights into fundamental particles and the structure of space-time but is also vital for modern technological systems that rely on accurate timekeeping and navigation.