Question

What is the value of \(\gamma\) for a particle moving at a speed of \(0.8 c ?\)

Short answer

Answer: The Lorentz factor for a particle moving at a speed of 0.8c is approximately 1.67.

Step-by-step solution

Understand the Lorentz Factor Formula

The Lorentz factor, denoted by \(\gamma\), is a fundamental concept in special relativity. It relates the time, length, and mass of an object moving at relativistic speeds to those measured in its rest frame. The formula to calculate the Lorentz factor is: $$\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$ Where: - \(v\) is the speed of the particle - \(c\) is the speed of light

Insert the Given Values

We are given that the particle is moving at a speed of \(0.8c\). To find the value of \(\gamma\), we'll substitute the given values into the formula: $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.8c)^2}{c^2}}}$$

Simplify the Expression

Now, let's simplify the expression by squaring the speed of the particle, dividing by the square of the speed of light, and subtracting the result from 1: $$\gamma = \frac{1}{\sqrt{1 - \frac{(0.64c^2)}{c^2}}}$$

Cancel out the Speed of Light

The \(c^2\) in the numerator and denominator will cancel each other out, which leaves us with: $$\gamma = \frac{1}{\sqrt{1 - 0.64}}$$

Solve for Gamma

Finally, we can calculate the value of \(\gamma\) by finding the square root: $$\gamma = \frac{1}{\sqrt{0.36}}$$ $$\gamma = \frac{1}{0.6}$$ $$\gamma \approx 1.67$$ Therefore, the value of \(\gamma\) for a particle moving at a speed of \(0.8c\) is approximately 1.67.

Key concepts

Lorentz Factor

The Lorentz factor, often represented by the Greek letter \( \gamma \), plays a crucial role in Albert Einstein's theory of special relativity. It helps us describe how much time, length, and mass change for objects moving at high speeds, close to the speed of light. The formula for the Lorentz factor is:\[\gamma = \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}\]Here, \( v \) is the speed of the moving object, and \( c \) is the speed of light. This factor becomes significant when an object's speed approaches the speed of light, as it leads to noticeable effects of time dilation (moving clocks tick more slowly) and length contraction (objects appear shorter in the direction of movement).

For example, in the provided exercise, we calculated the Lorentz factor for a particle moving at 80% of the speed of light, resulting in a value of \( \gamma \approx 1.67 \). This indicates the amount by which the time, length, and relativistic mass are altered from what they are in the rest state.

Relativistic Speeds

Relativistic speeds refer to velocities that are a significant fraction of the speed of light. When an object moves very fast, close to the speed of light, we must use the principles of special relativity to describe its motion accurately. At these speeds, everyday Newtonian physics doesn't suffice because time and space undergo distortions.

Some key points to understand about relativistic speeds:

• Time dilation: As an object moves at relativistic speeds, time starts to slow down relative to an observer at rest. This phenomenon is only noticeable at speeds that are a large portion of the speed of light.
• Length contraction: An object traveling at relativistic speeds will appear shorter in the direction of its motion when observed from a stationary frame of reference.
• Mass increase: As speed increases and approaches the speed of light, the relativistic mass of the object increases, making it harder to accelerate further.

Understanding relativistic speeds helps in studying particles in accelerators and also explains the behavior of cosmic phenomena, such as how particles travel across space.

Speed of Light

The speed of light, denoted by \( c \), is the fastest speed at which information or matter can travel through the universe. It is roughly \( 299,792,458 \) meters per second in a vacuum. This speed is not just a measure of how fast light travels, but a fundamental constant of nature that plays a vital role in the laws of physics.

The speed of light has several important implications:

• The ultimate speed limit: Nothing can travel faster than the speed of light in vacuum, according to the theory of relativity.
• Basis for special relativity: The constancy of the speed of light leads to the various relativistic effects, such as time dilation and length contraction.
• Energy-mass equivalence: The famous equation \( E = mc^2 \) shows the relationship between energy (E), mass (m), and the speed of light (c). This equation implies that even small amounts of mass can be converted into vast amounts of energy.

Grasping the concept of the speed of light is essential for comprehending the universe's physical laws, from intricate quantum mechanics to expansive cosmological phenomena.