Problem 86 - Algebra and Trigonometry

Question

Flight of a Rocket Suppose a rocket is fired upward from the surface of the earth with an initial velocity \(v\) (measured in meters per second). Then the maximum height \(h\) (in meters) reached by the rocket is given by the function $$ H(v)=\frac{R v^{2}}{2 g R-v^{2}} $$ where \(R=6.4 \times 10^{6} \mathrm{m}\) is the radius of the earth and \(g=9.8 \mathrm{m} / \mathrm{s}^{2}\) is the acceleration due to gravity. Use a graphing device to draw a graph of the function \(h .\) (Note that \(h\) and \(v\) must both be positive, so the viewing rectangle need not contain negative values.) What does the vertical asymptote represent physically?

Step-by-Step Solution

Answer

The vertical asymptote represents the escape velocity from Earth, where maximum height becomes infinite.

Step 1: Analyze the Function

The function given is \( H(v) = \frac{R v^{2}}{2 g R - v^{2}} \). Before drawing its graph, let's recognize its components. The function has a quadratic term \( v^2 \) in the numerator. The denominator \( 2 g R - v^2 \) suggests this will cause vertical asymptotes when it equals zero.

Step 2: Identify the Asymptote

To determine where the vertical asymptote occurs, set the denominator \( 2 g R - v^2 = 0 \). Solve this equation for \( v \):\[ 2 g R = v^2 \] \[ v = \sqrt{2 g R} \]. With \( R = 6.4 \times 10^6 \) and \( g = 9.8 \), calculate \( v \). This is the escape velocity.

Step 3: Calculate the Escape Velocity

Substitute the values for \( R \) and \( g \): \[ v = \sqrt{2 \times 9.8 \times 6.4 \times 10^6} \], \[ v \approx 11172 \, \text{m/s} \]. This is the velocity where the formula fails (vertical asymptote), meaning it's physically the minimum speed needed to escape Earth's gravity.

Step 4: Graph the Function

Using a graphing device, plot \( H(v) \) with \( v \) ranging from 0 to slightly less than \( 11172 \). Ensure your graph does not include \( v \) beyond this point, as the behavior changes with the vertical asymptote.

Step 5: Interpret the Vertical Asymptote

The vertical asymptote at \( v = 11172 \text{ m/s} \) means that as \( v \) approaches this velocity, the maximum height approaches infinity. Physically, this represents the escape velocity from Earth, where the rocket can theoretically reach an infinite height and escape Earth's gravitational pull.

Key Concepts

Escape Velocity

Escape velocity is the minimum speed needed for an object to break free from the gravitational attraction of a celestial body, such as Earth, without further propulsion. It is a fundamental concept in astrophysics and rocketry. For Earth, this speed is approximately 11,172 meters per second (or about 25,000 miles per hour).
At this velocity, an object can enter space completely overcoming the pull of Earth's gravity.

Several factors determine escape velocity:

• The mass of the celestial body
• The radius of the celestial body
• The gravitational constant

It’s important to note that escape velocity is different from orbital velocity, which is the speed needed to stay in orbit around a planet, rather than escaping it entirely.
In the given exercise, the escape velocity is the point where the mathematical function becomes undefined, indicating a transition from being under Earth's gravity to escaping it.

Graphing Functions

Graphing functions helps visualize mathematical relationships. In the exercise, the function describes the height a rocket reaches based on its initial velocity.
Understanding how to graph this function involves identifying key points, such as intercepts, asymptotes, and behavior at critical values.

Steps to graph a function like this include:

• Analyzing the function to determine where it is undefined. For the given function, this is the escape velocity.
• Calculating specific values to understand the range and end behavior of the graph.
• Using a calculator or graphing software for complex functions to visually see trends and patterns.

Graphing not only helps in visualizing the data, but it also allows for deeper insights into the relationships and comparisons between different variables affecting the rocket’s height.

Rocket Motion

Rocket motion is governed by the principles of physics which include understanding how forces like gravity impact a rocket's trajectory. When a rocket is fired, it starts with a certain velocity which affects how high it can ascend.
This motion can be described mathematically and associated with the concept of kinetic energy. The energy given to the rocket must be sufficient to overcome the gravitational pull of Earth.

Key considerations in understanding rocket motion include:

• The initial velocity of the rocket
• The gravitational force acting on the rocket
• The mass of the rocket

The exercise's formula illustrates that the velocity significantly influences the maximum height reached by the rocket, bridging concepts from mechanics and real-world applications.

Acceleration Due to Gravity

Acceleration due to gravity, denoted as \( g \,\), is the acceleration experienced by an object due to Earth's gravitational pull. Its average value is approximately 9.8 meters per second squared (m/s²) near Earth's surface. It is a key component in calculations involving motion and energy.
Gravity is the force that attracts a body towards the center of the Earth, or any other planetary body.

The role of gravity in calculations:

• It affects the trajectory of projectiles.
• In the motion of planets and satellites.
• In determining escape velocity, as seen in the exercise.

Without accounting for gravity, predicting the motion and potential reach of objects such as rockets would be impossible. Throughout the exercise, gravity is the fundamental force that opposes the rocket’s ascent, determining how much energy is needed to achieve specific heights or even escape from the planet.