Problem 28 - University Physics with Modern Physics

Question

As you have seen, relativistic calculations usually involve the quantity \(\gamma .\) When \(\gamma\) is appreciably greater than \(1,\) we must use relativistic formulas instead of Newtonian ones. For what speed \(v\) (in terms of \(c )\) is the value of \(\gamma(\mathrm{a}) 1.0 \%\) greater than \(1 ;\) (b) 10\(\%\) greater than 1 ; (c) 100\(\%\) greater than 1\(?\)

Step-by-Step Solution

Answer

For (a) \( v/c \approx 0.140 \), for (b) \( v/c \approx 0.563 \), and for (c) \( v/c \approx 0.866 \).

Step 1: Define gamma (γ)

The Lorentz factor, \( \gamma \), is defined by the equation:\[\gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}\]where \( v \) is the velocity of the object, and \( c \) is the speed of light in a vacuum.

Step 2: Determine the condition for γ being 1% greater than 1

For part (a), we need \( \gamma = 1.01 \). So we set:\[1.01 = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}\]Rearrange this to find \( \frac{v}{c} \):\[\sqrt{1 - \left( \frac{v}{c} \right)^2} = \frac{1}{1.01}\]\[1 - \left( \frac{v}{c} \right)^2 = \left( \frac{1}{1.01} \right)^2\]\[\left( \frac{v}{c} \right)^2 = 1 - \left( \frac{1}{1.01} \right)^2\]\[\frac{v}{c} = \sqrt{1 - \left( \frac{1}{1.01} \right)^2}\]

Step 3: Calculate the speed for part (a)

Compute \( \frac{v}{c} \) using the equation from Step 2:\[\frac{v}{c} \approx \sqrt{1 - \left( \frac{1}{1.01} \right)^2} \approx \sqrt{1 - 0.9802^2} \approx \sqrt{0.0198} \approx 0.140\]Therefore, for \( \gamma = 1.01 \), \( \frac{v}{c} \approx 0.140 \).

Step 4: Determine the condition for γ being 10% greater than 1

For part (b), we need \( \gamma = 1.10 \). Set:\[1.10 = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}\]Following similar steps as in Step 2:\[\left( \frac{v}{c} \right)^2 = 1 - \left( \frac{1}{1.10} \right)^2\]\[\frac{v}{c} = \sqrt{1 - \left( \frac{1}{1.10} \right)^2}\]

Step 5: Calculate the speed for part (b)

Compute \( \frac{v}{c} \) for a \( \gamma \) of 1.10:\[\frac{v}{c} \approx \sqrt{1 - \left( \frac{1}{1.10} \right)^2} \approx \sqrt{1 - 0.8264^2} \approx \sqrt{0.3175} \approx 0.563\]Thus, for \( \gamma = 1.10 \), \( \frac{v}{c} \approx 0.563 \).

Step 6: Determine the condition for γ being 100% greater than 1

For part (c), we need \( \gamma = 2.00 \), meaning it's 100% greater than 1. Set:\[2.00 = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}}\]So:\[\left( \frac{v}{c} \right)^2 = 1 - \left( \frac{1}{2.00} \right)^2\]\[\frac{v}{c} = \sqrt{1 - \left( \frac{1}{2.00} \right)^2}\]

Step 7: Calculate the speed for part (c)

Compute \( \frac{v}{c} \) for a \( \gamma \) of 2.00:\[\frac{v}{c} \approx \sqrt{1 - \left( \frac{1}{2.00} \right)^2} \approx \sqrt{1 - 0.25} \approx \sqrt{0.75} \approx 0.866\]Therefore, for \( \gamma = 2.00 \), \( \frac{v}{c} \approx 0.866 \).

Key Concepts

Lorentz Factor

In the realm of relativistic physics, the Lorentz factor, denoted by \( \gamma \), plays a crucial role in our understanding of how objects behave at high velocities. It emerges from Einstein's theory of relativity and accounts for the effects experienced as an object approaches the speed of light. The Lorentz factor is given by the equation:\[ \gamma = \frac{1}{\sqrt{1 - \left( \frac{v}{c} \right)^2}} \]where:

• \( v \) is the velocity of the object.
• \( c \) is the speed of light in a vacuum.

As the velocity \( v \) of an object gets closer to the speed of light \( c \), the denominator of the equation approaches zero, causing \( \gamma \) to increase rapidly. This factor affects time dilation, length contraction, and mass, all of which become significant in high-speed environments. Practically, if \( \gamma \) is notably greater than 1, Newtonian mechanics no longer accurately predict phenomena, underscoring the necessity of relativistic calculations.

Speed of Light

The speed of light, represented as \( c \), is fundamental to both physics and our understanding of the universe. It is a constant, approximately equal to \( 299,792,458 \) meters per second in a vacuum. This speed is not just a measure of how fast light travels but also a cosmic speed limit that nothing with mass can exceed. In the context of relativistic physics, the speed of light governs how energy, mass, and velocity interrelate.Some key points about the speed of light include:

• It is the maximum speed at which information and matter can travel.
• Light always travels at this speed in a vacuum, but it can be slower in other mediums like water or glass due to refraction.
• The constancy of the speed of light is a principle axiom in Einstein's theory of relativity, affecting how we perceive time and space.
• When performing relativity calculations, \( c \) serves as a benchmark for assessing velocities, such as when comparing \( v \) to \( c \) in the Lorentz factor.

Recognizing the importance of the speed of light helps in understanding how the universe behaves under the influence of relativistic effects.

Relativity Calculations

Relativity calculations are a cornerstone of understanding the behavior of objects moving at significant fractions of the speed of light. These calculations fundamentally alter our classical intuitions about speed, distance, and time.

Engaging with the Lorentz Factor

The use of the Lorentz factor \( \gamma \) is critical in these calculations. When \( \gamma \) is just slightly over 1, such as in the given task where we explore values of 1.01, 1.10, and 2.00, it indicates velocities becoming relativistically significant:

• For \( \gamma = 1.01 \), \( v/c \approx 0.140 \), a speed where relativistic effects begin to show subtle impact.
• For \( \gamma = 1.10 \), \( v/c \approx 0.563 \), showing more pronounced relativistic behavior.
• For \( \gamma = 2.00 \), \( v/c \approx 0.866 \), highlighting dominant relativistic impacts.

Consequences of Relativity

As velocities approach significant fractions of \( c \), classical physics no longer provides accurate predictions:

• Time dilation becomes apparent, meaning time moves slower for the moving object than for an observer at rest.
• Length contraction occurs, leading to the shortening of the object in the direction of motion as viewed by a stationary observer.
• Mass increases, affecting the energy dynamics of moving objects according to \( E = mc^2 \).

By understanding these calculations, students grasp how relative motion alters fundamental physical properties, reshaping our understanding of reality at high speeds.