Problem 16 - Thomas Calculus

Question

In Exercises 16 and \(17,\) two planets, planet \(A\) and planet \(B\) , are orbiting their sun in circular orbits with \(A\) being the inner planet and \(B\) being farther away from the sun. Suppose the positions of \(A\) and \(B\) at time \(t\) are $$ \mathbf{r}_{A}(t)=2 \cos (2 \pi t) \mathbf{i}+2 \sin (2 \pi t) \mathbf{j} $$ and $$ \mathbf{r}_{B}(t)=3 \cos (\pi t) \mathbf{i}+3 \sin (\pi t) \mathbf{j} $$ respectively, where the sun is assumed to be located at the origin and distance is measured in astronomical units. (Notice that planet \(A\) moves faster than planet \(B . )\) The people on planet \(A\) regard their planet, not the sun, as the center of their planetary system (their solar system). Using planet \(A\) as the origin of a new coordinate system, give parametric equations for the location of planet \(B\) at time \(t .\) Write your answer in terms of \(\cos (\pi t)\) and \(\sin (\pi t) .\)

Step-by-Step Solution

Answer

The location of planet B relative to planet A is: \( x(t) = 3 \cos (\pi t) - 2 \cos (2 \pi t) \), \( y(t) = 3 \sin (\pi t) - 2 \sin (2 \pi t) \).

Step 1: Understand the Problem

We are given the positions of two planets, A and B, orbiting a sun. The position of planet A is given by \( \mathbf{r}_{A}(t) = 2 \cos (2 \pi t) \mathbf{i} + 2 \sin (2 \pi t) \mathbf{j} \), and the position of planet B is given by \( \mathbf{r}_{B}(t) = 3 \cos (\pi t) \mathbf{i} + 3 \sin (\pi t) \mathbf{j} \). We are tasked with rewriting the position of planet B using planet A as the origin of the coordinate system.

Step 2: Set Up New Origin

To use planet A as the origin, we need to find the position of planet B relative to planet A. This involves subtracting the position vector of planet A from that of planet B, hence, \( \mathbf{r}_{B}^{\text{new}}(t) = \mathbf{r}_{B}(t) - \mathbf{r}_{A}(t) \).

Step 3: Calculate New Position of Planet B

Substitute the given position vectors into the equation from Step 2: \[ \mathbf{r}_{B}^{\text{new}}(t) = (3 \cos (\pi t) \mathbf{i} + 3 \sin (\pi t) \mathbf{j}) - (2 \cos (2 \pi t) \mathbf{i} + 2 \sin (2 \pi t) \mathbf{j}) \]. Simplifying, we obtain \( \mathbf{r}_{B}^{\text{new}}(t) = (3 \cos (\pi t) - 2 \cos (2 \pi t)) \mathbf{i} + (3 \sin (\pi t) - 2 \sin (2 \pi t)) \mathbf{j} \).

Step 4: Verify and Simplify the Result

Double-check the calculations for any simplification errors. The parametric equations for the new coordinates of planet B are: \( x(t) = 3 \cos (\pi t) - 2 \cos (2 \pi t) \) and \( y(t) = 3 \sin (\pi t) - 2 \sin (2 \pi t) \). These express the position of planet B when planet A is considered the origin.

Key Concepts

Circular Orbits

Circular orbits occur when an object moves around a center point in a circular path. This is based on gravitational attraction that keeps the orbiting body following a continuous route without deviating outwards or collapsing inwards.
In the exercise, both planets A and B orbit their sun in perfect circles. The equations provide the circular path these planets travel around their sun. Planet A has an orbit with a radius of 2 astronomical units (AU), while planet B has an orbit with a radius of 3 AU. To understand these circular orbits, consider them as continuous sequences of points in space given by their position vectors. The simplicity of circular orbits allows for straightforward calculations in terms of parametric equations.

Position Vectors

Position vectors are mathematical tools that give a location of a point in space relative to a fixed origin.In this exercise, these vectors are used to describe the positions of planets as they travel around their sun.
The position of a point in a 2D plane can be expressed as a combination of i-components and j-components.Planet A's position vector is given as \( \mathbf{r}_A(t) = 2 \cos(2\pi t) \mathbf{i} + 2 \sin(2\pi t) \mathbf{j} \).For planet B, the position vector is \( \mathbf{r}_B(t) = 3 \cos(\pi t) \mathbf{i} + 3 \sin(\pi t) \mathbf{j} \).The position vectors not only provide the coordinates of planets A and B at time \(t\) but also indicate the radii of their orbits.These vectors are essential for moving to a new coordinate system based on planet A as described later.

Relative Motion

Relative motion is important when analyzing how different observers perceive the movement of objects.This concept helps us understand how one object is moving in relation to another.The people on planet A, for example, consider their own planet as the central point of their observation.
In the exercise, this means we need to calculate the position of planet B relative to planet A.This is achieved by subtracting the position vector of planet A from the position vector of planet B.The equation becomes:\[ \mathbf{r}_{B}^{\text{new}}(t) = \mathbf{r}_{B}(t) - \mathbf{r}_{A}(t) \]These calculations help in understanding the relative position of one planet with respect to another as they follow their circular paths around the sun.

Coordinate System

The coordinate system serves as a set framework for identifying locations in space with respect to an origin.In this exercise, we examine how the coordinate system shifts when using planet A as the new reference point.
Usually, the sun is at the origin of the coordinate system, but residents of planet A use their own planet as the origin.To find planet B's position in this new system, we derive new parametric equations:\[ x(t) = 3 \cos(\pi t) - 2 \cos(2\pi t) \]\[ y(t) = 3 \sin(\pi t) - 2 \sin(2\pi t) \]These new equations reflect the altered perspective of planet A's inhabitants, aligning the coordinate system with their vantage point.Such shifts in the coordinate system are vital for understanding relative dynamics in various contexts, from planetary to engineering problems.