Problem 67 - Single Variable Calculus

Question

Mass of a Moving Particle The mass \(m\) of a particle moving at a speed \(v\) is related to its rest mass \(m_{0}\) by the equation $$ m=\frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}} $$ where \(c\), a constant, is the speed of light. Show that $$ \lim _{v \rightarrow c^{-}} \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\infty $$ thus proving that the line \(v=c\) is a vertical asymptote of the graph of \(m\) versus \(v .\) Make a sketch of the graph of \(m\) as a function of \(v\).

Step-by-Step Solution

Answer

In summary, as the speed \(v\) of a particle approaches the speed of light \(c^-\), its mass \(m\) approaches infinity, as proven by the limit: $$ \lim _{v \rightarrow c^{-}} \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\infty $$ The graph of the mass of the particle as a function of speed is characterized by: - Starting at \(m=m_0\) for \(v=0\), and increasing as \(v\) increases. - Having a vertical asymptote at \(v=c\), where the mass approaches infinity. This behavior highlights the fact that a particle cannot reach the speed of light, as its mass would become infinite.

Step 1: Evaluate the given limit as \(v \rightarrow c^-\)

To show that the given limit is infinity, we need to evaluate the limit: $$ \lim_{v \rightarrow c^-} \frac{m_{0}}{\sqrt{1-\frac{v^2}{c^2}}} $$ Let's analyze the denominator as \(v\) approaches the speed of light, \(c^-\): $$ \sqrt{1-\frac{v^2}{c^2}} $$ As \(v\) gets closer to \(c\), the fraction \(\frac{v^2}{c^2}\) approaches 1, so \(1-\frac{v^2}{c^2}\) approaches 0.

Step 2: Determine behaviour of the limit

Now that we know the denominator approaches 0 as \(v \rightarrow c^{-}\), we can reason about the limit. Since \(m_{0}\) is a constant and the denominator approaches 0, the whole expression: $$ \frac{m_{0}}{\sqrt{1-\frac{v^2}{c^2}}} $$ will approach infinity, as dividing a constant by a very small number tends to produce a very large number. Thus, $$ \lim _{v \rightarrow c^{-}} \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\infty $$

Step 3: Graphing the function

To sketch the graph of the mass \(m\) of the particle as a function of the speed \(v\), we can use the fact that we have just proven that the limit is infinity as \(v\) approaches \(c^-\). We know that, for low-speeds (as \(v \rightarrow 0\)), the denominator \(\sqrt{1-\frac{v^2}{c^2}}\) is equal to 1, and the mass of the particle \(m\) is equal to its rest mass \(m_0\). As \(v\) gets larger and approaches \(c\), the denominator approaches 0, and the mass of the particle approaches infinity. This means that the graph of the function will have a vertical asymptote at \(v = c\). The sketch of the graph will look like this: - The horizontal axis (x-axis) represents the speed \(v\). - The vertical axis (y-axis) represents the mass \(m\). - The function starts at \(m=m_0\) for \(v=0\) and increases as \(v\) increases. - At \(v=c\), there is a vertical asymptote, meaning the graph goes to infinity, and the particle's mass approaches infinity. With this understanding of the function, we can now sketch the graph of \(m\) as a function of \(v\).

Key Concepts

Limits in Calculus

In calculus, limits help us understand the behavior of functions as they approach a certain point. For instance, we often need to determine what happens to a function's value as the input gets very close to a specific number, or even to infinity. A limit like
\[\begin{equation}\lim_{x \to a} f(x) \end{equation}\],
is read as 'the limit of function f of x as x approaches a.' In simple terms, it represents the value that f(x) gets closer to as x gets closer to a. When limits yield infinity, as we see in the mass-speed relation equation for a particle approaching the speed of light, it represents a behavior without bound, growing larger and larger. Understanding limits is crucial for grasing concepts across all fields that use calculus, from physics to engineering.

Special Relativity

Special relativity is a theory proposed by Albert Einstein that fundamentally changed our understanding of space and time. One of the key principles is that the speed of light in a vacuum, denoted as c, is the same for all observers, regardless of their motion. Another striking result from this theory is the mass-speed relation of particles. The closer the speed of a particle gets to c, the greater its mass becomes. Thus, Einstein's equation \[\begin{equation}m = \frac{m_0}{\sqrt{1-\frac{v^2}{c^2}}} \end{equation}\], encapsulates a profound insight from special relativity: as an object's speed approaches c, its mass theoretically approaches infinity, preventing it from reaching or surpassing the speed of light.

Vertical Asymptotes

In the world of function graph sketching, a vertical asymptote is a line that a graph approaches but never touches or crosses. It occurs where the function moves towards infinite values as the input approaches a particular point. The equation discussed in the exercise \[\begin{equation}\lim _{v \rightarrow c^{-}} \frac{m_{0}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}=\infty \end{equation}\]illustrates such a behavior, indicating that the graph of the mass as a function of its speed has a vertical line at v=c that the graph will never intersect. Understanding where these asymptotes occur and why is essential in mathematical analysis and helps describe the limitations and behavior of functions in a visual format.

Function Graph Sketching

Graphing functions is a vital tool in mathematics for visualizing the relationship between variables. Sketching a graph involves plotting how the value of one variable changes in relation to another. In this context, we sketched the mass m as a function of speed v for a moving particle under special relativity principles. Here are the steps typically involved in function graph sketching:

• Determine the behavior of the function at critical points, including where it equals zero, and where it approaches infinity (vertical asymptotes).
• Analyze the slope or rate of change of the function — does it increase or decrease as the variable changes?
• Note any symmetry or repeating patterns that simplify the sketching process.
• Use these observations to plot the function on a graph, marking critical points and asymptotes, and then connect the dots smoothly to reveal the overall shape of the graph.

In our specific exercise, the graph starts with the mass equal to its rest mass at zero velocity and increases without bound as velocity approaches the speed of light, resulting in a curve that gets steeper and steeper, approaching a vertical line at v=c.