Problem 41 - Elementary Algebra

Question

Use numerical evaluation on the equations. Astronomy (Kepler's law of planetary motion) \(P^{2}=k a^{3} .\) Find \(P^{2}\) if \(k=4\) and \(a=5.1\)

Step-by-Step Solution

Answer

Answer: The value of \(P^2\) is approximately \(530.604\).

Step 1: Recall the equation for Kepler's law of planetary motion

The equation for Kepler's law of planetary motion is given as follows: \(P^2 = k \cdot a^3\) where: - \(P\) is the period of the planetary motion, - \(k\) is a constant, - \(a\) is the semi-major axis of the orbit. Our goal is to find the value of \(P^2\). We are given the values \(k = 4\) and \(a = 5.1\).

Step 2: Substitute the given values into the equation

Now, we will substitute the given values, \(k= 4\) and \(a = 5.1\), into the equation: \(P^2 = 4 \cdot (5.1)^3\)

Step 3: Perform numerical evaluation

Next, we will evaluate the expression by performing the arithmetic operations: \(P^2 = 4 \cdot (132.651) = 530.604\) Therefore, the value of \(P^2\) is approximately \(530.604\).

Key Concepts

Planetary Motion

Planetary motion refers to the way planets move in their orbits around a star, like the sun. It is based on Kepler's laws, which describe how these celestial objects travel through space. Kepler's first law states that planets orbit in ellipses, with the star at one focal point. Kepler's second law, often called the law of areas, suggests that a line joining a planet and its star sweeps out equal areas during equal intervals of time. This means planets move faster when closer to the star and slower when farther away.
Kepler's third law is the focus of our exercise. It reveals a relationship between the time a planet takes to complete an orbit, called the period, and the size of its orbit, represented by the semi-major axis. The mathematical expression is given as \( P^2 = k \cdot a^3 \), where \( P \) is the orbital period, \( a \) is the semi-major axis, and \( k \) is a constant. This law helps astronomers predict planet positions and understand their movements in the vast cosmic dance.

Numerical Evaluation

Numerical evaluation is a mathematical process used to find approximate values by computing an equation with given numbers. In the context of our exercise, we used numerical evaluation to find \( P^2 \) from Kepler's third law. This involved substituting specific values for \( k \) and \( a \) into the equation.
To numerically evaluate \( P^2 = k \cdot a^3 \), we followed these steps:

• Inserted the given constant \( k = 4 \) and the semi-major axis \( a = 5.1 \).
• Calculated \( a^3 \) by raising 5.1 to the power of 3, resulting in 132.651.
• Multiplied 132.651 by 4, resulting in 530.604.

This straightforward process of replacing variables with actual numerical values gives us a concrete understanding of planetary motion and how period squared relates to the orbit's size.

Astronomy

Astronomy is the scientific study of celestial bodies like stars, planets, comets, and galaxies, as well as phenomena that occur outside Earth's atmosphere. It's a field that brings the wonders of the universe to our understanding through observation and theoretical modeling.
Through astronomy, scientists have uncovered amazing facts about our solar system and beyond. Kepler's laws of planetary motion are a perfect example of foundational principles that emerged from astronomical observations. These laws not only explain how planets move but have also allowed astronomers to predict celestial events and discover exoplanets orbiting distant stars.
By using mathematics and scientific principles, astronomers aim to answer big questions about the beginnings of the universe and the potential for life beyond Earth. Astronomy also impacts technologies we use daily, including satellites and GPS systems. The continuous study of the heavens drives both scientific discoveries and technological innovations.