Problem 49 - Calculus

Question

When an object of mass \(m\) moves with a velocity \(v\) that is small compared to the velocity of light, \(c,\) its energy is given approximately by $$E \approx \frac{1}{2} m v^{2}$$ If \(v\) is comparable in size to \(c,\) then the energy must be computed by the exact formula $$E=m c^{2}\left(\frac{1}{\sqrt{1-v^{2} / c^{2}}}-1\right)$$ (a) Plot a graph of both functions for \(E\) against \(v\) for \(0 \leq v \leq 5 \cdot 10^{8}\) and \(0 \leq E \leq 5 \cdot 10^{17} .\) Take \(m=1 \mathrm{kg}\) and \(c=3 \cdot 10^{8} \mathrm{m} / \mathrm{sec} .\) Explain how you can predict from the exact formula the position of the vertical asymptote. (b) What do the graphs tell you about the approximation? For what values of \(v\) does the first formula give a good approximation to \(E ?\)

Step-by-Step Solution

Answer

The approximation is accurate for velocities much smaller than the speed of light. For \(v < 0.1c\), the approximate formula is a good approximation.

Step 1: Understand the Problem

We have two expressions to calculate energy based on the velocity of an object: an approximate formula for low velocity and an exact formula for velocities comparable to the speed of light. Our tasks are to plot these formulas over a specified range of velocities and energies and to analyze their accuracy and behavior.

Step 2: Setup Equations for Plotting

First, set up both energy functions with given parameters: - Approximate: \(E_\text{approx} = \frac{1}{2} m v^2\) - Exact: \(E_\text{exact} = m c^2 \left( \frac{1}{\sqrt{1-v^2/c^2}} - 1 \right)\).Use \(m = 1\, \text{kg}\), \(c = 3 \times 10^8\, \text{m/s}\).

Step 3: Calculate Energy for Approximate Formula

For values of \(v\) from 0 to \(5 \times 10^8\), compute energies using the approximate formula: \[ E_\text{approx} = \frac{1}{2} \times 1 \times v^2 = 0.5 v^2 \]

Step 4: Calculate Energy for Exact Formula

For the same range of velocities, compute energies using the exact formula: \[ E_\text{exact} = 1 \times (3 \times 10^8)^2 \left( \frac{1}{\sqrt{1-v^2/(3 \times 10^8)^2}} - 1 \right) \]

Step 5: Plot the Energy Functions

Plot both results on the same graph: velocity \(v\) on the x-axis and energy \(E\) on the y-axis. Both functions range over \(0 \leq v \leq 5 \times 10^8\) and \(0 \leq E \leq 5 \times 10^{17}\).

Step 6: Analyze the Vertical Asymptote Behaviour

From the exact formula, note that a vertical asymptote occurs as the speed \(v\) approaches the speed of light \(c\), due to the denominator approaching zero: \[ \sqrt{1-v^2 / c^2} \to 0 \text{ as } v \to c \]. Therefore, a vertical asymptote is present at \(v = c = 3 \times 10^8\).

Step 7: Interpretation of Graphs

Inspect the plotted graphs. The approximate formula closely follows the exact one at lower velocities \(v \ll c\), diverging notably as \(v\) approaches \(c\). The approximation is good when \(v\) is significantly less than \(c\), particularly noticeable when \(v < 0.1c\).

Key Concepts

Energy Approximation

In physics, the concept of energy approximation is crucial when dealing with objects moving at relatively low velocities. The approximation is simpler to compute and only requires the mass and velocity of the object. When an object's velocity (\(v\)) is significantly lower than the speed of light (\(c = 3 \times 10^8\, \text{m/s}\)), its kinetic energy can be approximated by the formula \(E \approx \frac{1}{2} m v^2\). This formula is derived from classical physics and assumes that relativistic effects are negligible at such speeds.

Understanding this approximation helps in identifying when using this simplified expression is appropriate. It is valid for many everyday scenarios, such as calculating the energy of a car driving on a highway or a ball being thrown. However, as velocities rise toward a significant fraction of the speed of light, this approximation becomes less accurate and must be replaced with relativistic calculations for valid results.

Velocity and Energy Relationship

The velocity and energy relationship in the context of relativistic physics involves understanding how energy calculations change when dealing with velocities near the speed of light. As an object approaches these vast speeds, its energy no longer increases linearly but instead follows a more complex expression. This behavior is described by the exact energy formula: \[E = mc^2 \left( \frac{1}{\sqrt{1-v^2 / c^2}} - 1 \right)\]This formula accounts for the effects of special relativity, predicting that as \(v\) approaches \(c\), the energy required to continue accelerating the object increases dramatically.

For low velocities, the simpler energy approximation is very close to the exact relativistic formula. This is why the approximation remains a staple in calculations where the velocity is much less than the speed of light. However, as velocity increases, a divergence becomes evident: the exact formulation shows significantly higher energy values, reflecting the additional energy needed due to relativistic effects.

This understanding highlights an important principle: energy grows increasingly large as velocities approach light speed, which illustrates the core tenet of Einstein's theory of relativity that prohibits any massive object from reaching or exceeding the speed of light.

Vertical Asymptote Behavior

Vertical asymptote behavior is a key aspect of analyzing functions, like the relativistic energy equation. A vertical asymptote occurs in a graph where the function tends to infinity as the input values approach a certain point. For the exact energy formula \(E = mc^2 \left( \frac{1}{\sqrt{1-v^2 / c^2}} - 1 \right)\), there is a vertical asymptote as \(v\) approaches \(c\).

This phenomenon occurs because the denominator in the fraction under the square root, \(\sqrt{1-v^2 / c^2}\), approaches zero. As a result, the entire expression tends towards infinity, indicating a vertical asymptote at \(v = c\). This asymptote graphically represents the impracticality of reaching or surpassing the speed of light for any object with mass.

Understanding vertical asymptotes helps explain why energy expectations change dramatically at high velocities. They signify points where typical physical intuition breaks down, and relativistic effects dominate. The presence of these asymptotes in the mathematical expression reinforces the limitations imposed by the theory of relativity on the velocity of massive objects.