Problem 139 - Elementary and Intermediate Algebra

Question

Relativity. One concept of relativity theory is that an object moving past an observer at a speed near the speed of light appears to have a larger mass because of its motion. If the mass of the object is \(m_{0}\) when the object is at rest relative to the observer, its mass \(m\) will be given by the formula \(m=m_{0}\left(1-\frac{v^{2}}{c^{2}}\right)^{-1 / 2}\) when it is moving with speed \(v\) (in miles per second) past the observer. The variable \(c\) is the speed of light, \(186,000\) mi/sec. If a proton with a rest mass of 1 unit is accelerated by a nuclear accelerator to a speed of \(160,000 \mathrm{mi} / \mathrm{sec},\) what mass will the technicians observe it to have? Round to the nearest hundredth.

Step-by-Step Solution

Answer

The observed mass is approximately 1.96 units.

Step 1: Understand the Problem

We need to find the observed mass of a proton moving at a high speed. We are given the rest mass \( m_0 = 1 \) unit and speed \( v = 160,000 \) mi/sec. The speed of light is \( c = 186,000 \) mi/sec. Using the relativity formula, we will calculate the moving mass \( m \).

Step 2: Recall the Formula

The formula for the mass of an object moving at relativistic speeds is:\[m = m_0 \left(1-\frac{v^2}{c^2}\right)^{-1 / 2}\]where \( m_0 \) is the rest mass, \( v \) is the velocity of the object, and \( c \) is the speed of light.

Step 3: Substitute Given Values into the Formula

Substitute \( m_0 = 1 \), \( v = 160,000 \text{ mi/sec} \), and \( c = 186,000 \text{ mi/sec} \) into the formula:\[m = 1 \left(1-\frac{(160,000)^2}{(186,000)^2}\right)^{-1 / 2}\]

Step 4: Calculate the Fraction under the Square Root

First, calculate \( v^2 \):\[v^2 = (160,000)^2 = 25,600,000,000\]Next, calculate \( c^2 \):\[c^2 = (186,000)^2 = 34,596,000,000\]Now, find the ratio \( \frac{v^2}{c^2} \):\[\frac{v^2}{c^2} = \frac{25,600,000,000}{34,596,000,000} \approx 0.739\]

Step 5: Compute the Expression under the Square Root

Subtract the calculated ratio from 1:\[1 - \frac{v^2}{c^2} = 1 - 0.739 = 0.261\]

Step 6: Calculate the Observed Mass

Calculate the reciprocal square root:\[\left(0.261\right)^{-1/2} = \frac{1}{\sqrt{0.261}} \approx 1.956\]Thus, the observed mass \( m \) is:\[m = 1 \times 1.956 = 1.956\]

Step 7: Round the Result

Round \( 1.956 \) to the nearest hundredth:\[m \approx 1.96\]

Key Concepts

Mass-Energy Equivalence

Let's dive into one of the mind-blowing insights of Einstein's theory of relativity known as mass-energy equivalence. This concept is famously encapsulated in the formula \( E = mc^2 \). It shows that mass and energy are interchangeable. Essentially, it suggests that mass can be transformed into energy and vice versa. This is revolutionary because it implies that even a small amount of mass can be converted into a large amount of energy, due to the speed of light squared being a massive number.

In practical terms:

• Mass and energy are two sides of the same coin.
• Converting mass to energy helps us understand processes like nuclear fission and fusion.

To put it simply, if an object gains energy, it also gains mass. This relationship becomes incredibly significant at speeds near the speed of light, where the increase in energy from the object's motion is substantial.

Relativistic Mass

When discussing relativity, it's important to consider the concept of relativistic mass. This idea emerges when an object moves at a velocity close to the speed of light. As an object's speed increases, its mass appears to increase from the perspective of an observer.

Consider this:

• Rest mass \(m_0\) is the mass an object has when it is not moving relative to an observer.
• As an object moves faster, approaching the speed of light, its mass \(m\) increases and is given by: \[m = m_0 \left(1-\frac{v^2}{c^2}\right)^{-1/2}\]
• This increase is significant at speeds near the speed of light, making objects behave as if they are more massive than their rest mass.

The formula shows that as \(v\) approaches \(c\), the denominator approaches zero, making \(m\) theoretically infinite. It's a bit like the universe's speed limit, stamping a boundary on how fast objects with mass can travel.

Speed of Light

The speed of light, represented as \(c\), is one of the most fundamental constants in physics. It is approximately \(186,000\) miles per second (or \(299,792,458\) meters per second). This constant isn't just about how fast light travels but also acts as a speed limit for the universe.

Here's what makes the speed of light so crucial:

• No matter how fast an object moves, it cannot surpass the speed of light. This is because as an object accelerates toward the speed of light, its mass increases, requiring more and more energy to continue accelerating.
• Light speed interlinks space and time in the fabric we call spacetime, stretching and bending under gravitational forces and high velocities.
• The invariance of \(c\) underpins the theory of relativity, affecting how we measure time, distance, and the mass of objects moving at high speeds.

The notion that nothing can outpace light rests at the heart of understanding relativity, shaping how we perceive the universe.