Problem 52 - Precalculus

Question

52\. Allow the speed of light, \(c,\) to be equal to 1.0 . If the mass, \(m,\) is \(1,\) what occurs to \(m\) as \(v \rightarrow c ?\) Using the values listed in Table \(12.1,\) make a conjecture as to what the mass is as \(v\) approaches 1.00 . $$\begin{array}{|c|c|c|c|c|c|}\hline v & {0.5} & {0.9} & {0.95} & {0.99} & {0.999} & {0.99999} \\ \hline m & { 1.15} & {2.29} & {3.20} & {7.09} & {22.36} & {223.61} \\ \hline\end{array}$$

Step-by-Step Solution

Answer

As \(v\) approaches \(c = 1\), the mass \(m\) approaches infinity.

Step 1: Understand the concept of relativistic mass

When dealing with speeds close to the speed of light, relativistic effects become apparent. According to special relativity, the mass of an object increases with its velocity. The relationship is given by the formula \[ m(v) = \frac{m_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}} \]where \(m_0\) is the rest mass, \(v\) is the velocity of the object, and \(c\) is the speed of light.

Step 2: Set up the scenario with given values

In this scenario, the speed of light \(c\) is set to 1.0, simplifying the equation to: \[ m(v) = \frac{m_0}{\sqrt{1 - v^2}} \]with \(m_0 = 1\). The table provides different velocities \(v\) and their respective calculated masses.

Step 3: Analyze the given values from the table

Look at the table with values of \(v\) and corresponding \(m\). As \(v\) gets closer to \(1\), \(m\) increases significantly. For example:- \(v = 0.5\), \(m = 1.15\)- \(v = 0.9\), \(m = 2.29\)- \(v = 0.95\), \(m = 3.20\)- \(v = 0.99\), \(m = 7.09\)- \(v = 0.999\), \(m = 22.36\)- \(v = 0.99999\), \(m = 223.61\)This data shows how mass increases dramatically as \(v\) approaches \(1\).

Step 4: Make a conjecture for when \(v\) approaches \(c\)

As \(v\) approaches \(c = 1\), the denominator in the mass formula \(\sqrt{1 - v^2}\) approaches zero, which causes \(m\) to become infinite. This implies that as an object's velocity approaches the speed of light, its mass theoretically becomes infinite.

Key Concepts

Special Relativity

Special relativity is a theory proposed by Albert Einstein that revolutionized our understanding of physics. It primarily deals with the physics of high-speed objects and introduces a new way to perceive time and space. The most famous equation from this theory, \(E=mc^2\), implies that mass and energy are equivalent.
This means that energy can be converted into mass and vice versa. This theory essentially tells us that the laws of physics are the same for all non-accelerating observers and that the speed of light is constant in any frame of reference.

• Impacts the understanding of mass, time, and space.
• Shows how time can dilate and lengths contract as objects move close to the speed of light.

These ideas are crucial for understanding how relativistic mass changes as an object's velocity changes.

Velocity

Velocity refers to the speed of an object in a particular direction. It is a vector quantity, which means it has both magnitude (speed) and direction.
In the context of special relativity, velocity plays a critical role in determining relativistic effects. As velocity approaches the speed of light, \(c\), significant changes happen to various properties, like time, length, and mass.
The term "relativistic velocity" implies that certain physical quantities, like mass, behave differently than they do at normal speeds. To calculate relativistic mass as velocity increases, the formula \(m(v) = \frac{m_0}{\sqrt{1 - \left(\frac{v}{c}\right)^2}}\) is used.

• Makes mass increase as speed nears the speed of light.
• Provides insight into how objects behave at high speeds.

This concept is key in understanding why an object's mass appears to increase with its speed.

Rest Mass

Rest mass, denoted as \(m_0\), is the mass of an object as measured when it is at rest. This is also known as invariant mass because it remains constant regardless of the object's velocity.
In the context of special relativity, rest mass is the "true" mass of an object devoid of relativistic effects.
However, when an object moves at speeds approaching the speed of light, we have to account for relativistic mass instead. Rest mass is used in the formula for calculating relativistic mass in moving objects.

• Acts as a baseline for determining changes in mass at high velocities.
• Doesn't change with velocity but contributes to relativistic measurements.

This helps clarify how mass varies with different speeds, tying into the equations provided in the solutions.

Speed of Light

The speed of light, denoted as \(c\), is approximately \(299,792,458\) meters per second. It is fundamental in physics because it is the maximum speed at which information or matter can travel through space.
Special relativity posits that nothing can move faster than the speed of light.
In this exercise, setting \(c = 1.0\) simplifies calculations and emphasizes the relation between speed and mass.

• Defines the upper limit for speed in the universe.
• Essential for understanding relativistic mass changes as shown in the exercise.

This constant is central to calculations involving relativistic effects, like infinite mass when velocity reaches \(c\).