Problem 79 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 79-80: During a nuclear explosion, a fireball will be produced having a maximum volume \(V_{0}\). For temperatures below \(2000 \mathrm{~K}\) and a given explosive force, the volume \(V\) of the fireball \(t\) seconds after the explosion can be estimated using the given formula. (Note that the kelvin is abbreviated as \(\mathrm{K}\), not \({ }^{\circ} \mathrm{K}_{.}\)) Approximate \(t\) when \(V\) is \(95 \%\) of \(V_{0}\). \(\mathrm{~V} / V_{0}=0.8197+0.007752 t+0.0000281 t^{2}\) (20-kiloton explosion)

Step-by-Step Solution

Answer

The time \( t \) when \( V \) is 95% of \( V_0 \) can be solved using the quadratic formula.

Step 1: Understand the Problem

We are given a formula \( \frac{V}{V_0}=0.8197+0.007752 t+0.0000281 t^{2} \), which models the volume \( V \) of a nuclear fireball produced by a 20-kiloton explosion in terms of time \( t \). We need to determine the time \( t \) when \( V \) is 95% of \( V_0 \), which translates to \( \frac{V}{V_0} = 0.95 \).

Step 2: Set the Problem Equation

To find \( t \) when \( V \) is 95% of \( V_0 \), set the equation \( 0.8197 + 0.007752t + 0.0000281t^2 = 0.95 \).

Step 3: Rearrange the Equation

Rearrange the equation to make it easier to solve: \( 0.0000281t^2 + 0.007752t + 0.8197 - 0.95 = 0 \). Simplify it to \( 0.0000281t^2 + 0.007752t - 0.1303 = 0 \).

Step 4: Apply the Quadratic Formula

Use the quadratic formula to solve for \( t \), where \( a = 0.0000281 \), \( b = 0.007752 \), and \( c = -0.1303 \). The formula is \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

Step 5: Calculate the Discriminant

Calculate the discriminant \( b^2 - 4ac \): \( (0.007752)^2 - 4 \times 0.0000281 \times (-0.1303) \). Compute this value.

Step 6: Simplify and Solve for t

Substitute the discriminant back into the quadratic formula and solve for the two possible values of \( t \). Choose the positive value since time cannot be negative.

Key Concepts

Nuclear Explosions

When a nuclear explosion occurs, it releases an enormous amount of energy. This release of energy is what makes it remarkably powerful and devastating. The energy comes from nuclear reactions, where the nucleus of an atom splits (fission) or combines (fusion) to release energy. These reactions release energy in the form of a large fireball, intense radiation, and shock waves.

• The fireball is incredibly hot, with temperatures reaching thousands of degrees Kelvin, and expands rapidly in the air.
• The explosive force comes from the rapid expansion caused by the immense heat, creating high-pressure waves that spread outward.
• Nuclear explosions are measured in terms of tons of TNT equivalent, with a 20-kiloton explosion being equivalent to 20,000 tons of TNT.

Understanding the scale and impact of nuclear explosions helps in analyzing the dynamics of the resulting fireball and the force it exerts on its environment.

Volume Estimation

Estimating the volume of a fireball in a nuclear explosion involves understanding how the fireball expands over time. The volume changes significantly in the first few seconds as the fireball reaches its maximum size.
In the provided problem, students must estimate the volume at a specific time after an explosion. The formula given models the volume in terms of the percentage of maximum volume over time. The challenge is to determine the time when this reaches 95% of its peak.

• To solve such a problem, we use a polynomial equation derived from measurements and observations.
• This polynomial typically relates the volume percentage to time using quadratic terms because the growth behavior can be parabolic in nature.
• Here, a quadratic equation is used because it conveniently expresses the relationship between time and fireball growth applicable under specified conditions.

Fireball Dynamics

Fireball dynamics refer to the behavior of the explosive fireball that emerges seconds after a nuclear detonation. The fireball's dynamics are primarily governed by the extreme temperatures and expansion rates right after the explosion.

• Initially, the fireball is prominently luminous and expands rapidly, driven by intense heat from nuclear reactions.
• The expansion rate is significant in the first few seconds, often following a predictable pattern allowing for mathematical modeling.
• Factors such as the explosive yield and atmospheric conditions influence the fireball's ultimate size and brightness.

In mathematical terms, analyzing fireball dynamics involves applying physics and mathematical equations that capture these rapid changes in size and temperature. This helps predict how long it will take to reach certain sizes or surface temperatures.

Explosive Force Analysis

Analyzing the explosive force of a nuclear detonation involves understanding the energy distribution during the explosion. The explosive force is a result of rapidly expanding gases and energy released from nuclear reactions.

• The pressure wave, or shockwave, created by this force can cause significant damage over vast areas.
• The power of the explosion (such as a 20-kiloton bomb) can be used to calculate its effects on buildings and structures in its vicinity.
• By studying the explosive force, analysts can determine the potential impact zone and necessary safety measures.

Such analysis helps in safeguarding environments and creating potential mitigation strategies. Understanding these concepts is crucial for disaster preparedness and reducing risks associated with nuclear explosions.