Question

A simplified model assumes the orbits of Earth and Mars around the Sun 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 \((3,0) ; \text { both orbit the Sun (at }(0,0))\) 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 Farth is a limacon (Exercise 91 ).

Short answer

Answer: The shape of the path of Mars relative to Earth is a limacon. This is because the equation for the radius r, in which r = (3 - 4cos(πt)), can be rewritten in the form of a limacon equation, r = a + bcos(θ), where a and b are constants and θ is the polar angle. In this case, a = 3 and b = -4, which confirms the shape of the path as a limacon.

Step-by-step solution

Analyzing the orbits and given positions

Given that the orbits of Earth and Mars are circular with radii of 2 and 3 respectively, Earth completes an orbit in one year while Mars takes two years. At \(t=0\), Earth is at \((2,0)\) and Mars is at \((3,0)\), both orbiting the Sun at \((0,0)\) in the counterclockwise direction.

Graph the parametric equations

The parametric equations provided for the position of Mars relative to Earth are given as $$ x = (3 - 4\cos(\pi t))\cos(\pi t) + 2, $$ and $$ y = (3 - 4\cos(\pi t))\sin(\pi t). $$ To graph the parametric equations, we can use a graphing software or plot points for different values of \(t\) in the interval \(0 \leq t \leq 2\). For each value of \(t\), calculate the corresponding \(x(t)\) and \(y(t)\), and then plot them on the Cartesian plane. The graph of these parametric equations will show the path of Mars relative to Earth over two years.

Explaining the limacon shape

The equation for the radius \(r\) is given as $$ r = (3 - 4\cos(\pi t)). $$ A limacon is a polar curve that has the equation $$ r = a + b\cos(\theta), $$ where \(a\) and \(b\) are constants, and \(\theta\) is the polar angle. We can rewrite the given equation for \(r\) using the polar angle \(\theta = \pi t\) to make the comparison with the limacon equation more apparent: $$ r = 3 - 4\cos(\theta). $$ Comparing the above equation with the equation of a limacon, we can see that \(a = 3\) and \(b = -4\). Since both \(a\) and \(b\) are constants, we can conclude that the path of Mars relative to Earth is a limacon.

Key concepts

Parametric Equations

Parametric equations are a set of equations that express a set of quantities as explicit functions of a number of independent parameters. These equations are particularly useful in tracking the path of moving objects. In our exercise, the path of Mars relative to Earth is represented by parametric equations.

• The equations for Mars' position in terms of time, \(t\), are:
  \[x = (3 - 4\cos(\pi t))\cos(\pi t) + 2,\]
  \[y = (3 - 4\cos(\pi t))\sin(\pi t).\]
• Each of these equations describes a coordinate (\(x\) or \(y\)) as a function of \(t\), which ranges from 0 to 2 in this scenario.
• To visualize these equations, you would calculate \(x\) and \(y\) for several values of \(t\), then plot these points on a graph.

This allows us to understand the trajectory Mars takes relative to Earth, and if you graph it, it will resemble a curve called a limacon.

Polar Coordinates

Polar coordinates offer a different way to express location in a plane compared to Cartesian coordinates, using a distance from a reference point and an angle from a reference direction. In this exercise, the polar coordinate aspect shines through in the expression of the path of Mars.

• Polar coordinates are often denoted as \((r, \theta)\), where \(r\) is the radial distance and \(\theta\) is the angular coordinate.
• The solution uses this concept by deriving \(r\) as a function of \(\theta = \pi t\), rewriting it as \[r = 3 - 4\cos(\theta).\]
• This conversion from a parametric to a polar form reveals the path's shape is a limacon.

These coordinates are widely used in fields dealing with circular or rotational motions, such as physics and engineering, because they simplify calculations involving angles and distances from a central point.

Orbital Mechanics

Orbital mechanics is the study of the motions of celestial bodies affected mainly by gravitational forces. The principles are applied here to understand the orbits of Earth and Mars.

• The Earth and Mars have been simplified in this problem to orbit the sun in circular paths with radii of 2 and 3 units, respectively.
• Earth completes an orbit in one year, while Mars takes two years, showing variations in orbital period and radius.
• The parametric equations used for illustrating Mars' position relative to Earth take into account these orbital mechanics.

Understanding these concepts helps us not only in plotting their paths but also in explaining the results, such as why the path of Mars relative to Earth takes the shape of a limacon. Such knowledge extends to practical applications in planning space missions or understanding satellite trajectories.