Problem 109 - Calculus Early Transcendentals

Question

A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. When \(t=0,\) Earth is at (2,0) and Mars is at (3,0) both orbit the Sun (at (0,0) ) in the counterclockwise direction. The position of Mars relative to Earth is given by the parametric equations \(x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t\) a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars relative to Earth is a limaçon (Exercise 89).

Step-by-Step Solution

Answer

Answer: The path of Mars relative to Earth is a limaçon because its radial distance function, given by \(r=(3-4 \cos \pi t)\), can be rewritten in the form of the polar equation for a limaçon, \(r=a + b\cos\theta\). In this case, we have \(a=3\) and \(b=-4\). By mapping the time \(t\) to the polar angle \(\theta = \pi t\), we can see that the path of Mars relative to Earth follows a limaçon curve.

Step 1: Understand the given information

Earth and Mars are orbiting the Sun in a counterclockwise direction. At time t=0, Earth is at the point (2,0), and Mars is at the point (3,0). Earth completes its orbit in 1 year, whereas Mars takes 2 years to complete its orbit. The position of Mars relative to Earth is given by the parametric equations: \(x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t\)

Step 2: Graph the parametric equations

We need to graph the given parametric equations for \(0 \leq t \leq 2\). We can do this by plotting a set of points \((x(t), y(t))\) for different values of \(t\) in increments of 0.1, for example. You can use software like Desmos, GeoGebra, or a graphing calculator to plot the points.

Step 3: Observe the path of Mars relative to Earth

After graphing the parametric equations, we can see that the path of Mars relative to Earth creates a closed curve. This curve resembles a limaçon, which is a curve defined by the polar equation \(r=a + b\cos\theta\) or \(r=a + b\sin\theta\) where \(a\) and \(b\) are constants.

Step 4: Explain the similarity to a limaçon

We are given \(r=(3-4 \cos \pi t)\), which is a function of the cosine wave. To compare it with the polar equation of the limaçon, we must find the mapping from \(t\) to the polar angle \(\theta\). Since Earth orbits the Sun in 1 year, and both Earth and Mars move counterclockwise, we can map the time \(t\) to the polar angle in the following way: \(\theta = \pi t\). Using this mapping, we can rewrite the given equation for \(r\) as \(r = 3 - 4\cos(\pi t) = 3 - 4\cos(\theta)\), which is a limaçon equation with \(a=3\) and \(b=-4\), in the form \(r=a + b\cos\theta\). Therefore, the path of Mars relative to Earth is indeed a limaçon.

Key Concepts

Limaçon

The term "limaçon" might sound exotic, but it simply refers to a special type of mathematical curve. Limaçons are part of a broader category called epicycloids, which are curves created by tracing points on a circle as it rolls around the outside of another circle.

Specifically, a limaçon can be described using polar coordinates with equations like \(r = a + b \cos \theta\) or \(r = a + b \sin \theta\), where \(a\) and \(b\) are constants. These curves can appear as loops, dimples, or even circles, depending on the values of \(a\) and \(b\).

• If \(|b| > a\), the limaçon has a loop.
• If \(a = |b|\), it resembles a cardioid.
• If \(|b| < a\), the limaçon appears dimpled.

In our exercise, the path of Mars relative to Earth forms a limaçon due to the equations involving cosine. By rearranging these equations, we see that they match the standard limaçon form, highlighting the fascinating geometry of planetary motion. Limaçons elegantly illustrate how mathematical expressions translate into beautiful and often surprising patterns in both mathematics and nature.

Orbital Mechanics

Orbital mechanics is the field of science that analyzes how objects like planets, moons, and satellites move in space. This science helps us predict how celestial bodies travel over time using laws first described by Johannes Kepler and Isaac Newton.

In our exercise, we deal with simplified yet fascinating aspects of orbital mechanics involving Earth and Mars. Both planets are modeled to move in circular orbits around the Sun. Mars takes twice as long to complete its orbit, so its path relative to Earth creates unique patterns captured by parametric equations.

• Earth completes an orbit in one year.
• Mars completes its orbit in two years, creating varied viewing angles from Earth.
• Parametric equations help visualize Mars' relative position during these motions.

By understanding how these parametric equations represent the orbit over time, one can begin to grasp the complex yet harmonious dance that planets undertake in their vast cosmic journey. This helps lay the groundwork for further studies in astronomy and space exploration.

Graphing

Graphing parametric equations is an essential skill for visualizing complex paths and curves. These equations describe multiple variables changing over time, resulting in paths different from simple linear or quadratic equations.

In our exercise, plotting the path of Mars relative to Earth requires creating a graph from the given parametric equations. Graphing in this context means calculating point values \((x(t), y(t))\) for different values of \(t\). When plotted, these points reveal the intricate shapes made by celestial mechanics.

• Choose a software tool like Desmos or GeoGebra for easy plotting.
• Calculate points for \(t\) values in small enough increments (e.g., 0.1) for smooth curves.
• Observe how the parametric form is crucial in capturing motion over time.

Modern graphing technologies allow us to visualize these shapes quickly, without laborious manual calculations. This process not only helps us understand mathematical functions but also offers insight into real-world phenomena like planetary motion. By connecting algebraic expressions with visual graphs, graphing bridges intuition with analytic reasoning.