Problem 50 - Algebra and Trigonometry

Question

Gravity Near the Moon We can use Newton's Law of Gravity to measure the gravitational attraction between the moon and an algebra student in a space ship located a distance \(x\) above the moon's surface: $$F(x)=\frac{350}{x^{2}}$$ Here \(F\) is measured in newtons \((\mathrm{N}),\) and \(x\) is measured in millions of meters. (a) Graph the function \(F\) for values of \(x\) between 0 and \(10 .\) (b) Use the graph to describe the behavior of the gravitational attraction \(F\) as the distance \(x\) increases.

Step-by-Step Solution

Answer

As distance \( x \) increases, gravitational attraction \( F(x) \) decreases rapidly.

Step 1: Understand the Function

The given function for gravitational force is \( F(x) = \frac{350}{x^2} \). The force \( F \) is a function of \( x \), where \( x \) represents the distance from the moon in millions of meters. This is an inverse square law, indicating that as \( x \) increases, \( F(x) \) decreases, reflecting the weaker gravitational attraction with increasing distance.

Step 2: Graph the Function

To graph \( F(x) = \frac{350}{x^2} \) for values between 0 and 10, we start by determining several values of \( x \) and calculating corresponding \( F(x) \) values. For example, when \( x = 1 \), \( F(1) = \frac{350}{1^2} = 350 \); when \( x = 2 \), \( F(2) = \frac{350}{2^2} = 87.5 \); and when \( x = 10 \), \( F(10) = \frac{350}{10^2} = 3.5 \). Plot these calculated points and draw a smooth curve through them to visualize how \( F(x) \) changes with \( x \).

Step 3: Describe the Graph's Behavior

The graph is a downward-sloping curve that approaches zero as \( x \) increases. At \( x = 0 \), the function approaches a value that would be undefined (x cannot actually be zero), indicating an asymptote there. As \( x \) increases from 1 to 10, \( F(x) \) decreases rapidly at first and then the rate of decrease slows, becoming almost flat.

Step 4: Analyze Influence of Distance on Gravity

As the distance \( x \) from the moon increases, the gravitational force \( F(x) \) diminishes at a decreasing rate. This means the force of attraction does not drop to zero but becomes significantly weaker as the distance elevates to 10 million meters and beyond. Such an inverse-square relation reveals that gravitational forces become less effective very quickly with increasing distance.

Key Concepts

Gravitational Force

Gravitational force is a fundamental interaction that attracts two objects with mass towards each other. In the context of Newton's Law of Gravity, this force is what keeps planets in orbit and governs the motion of celestial objects. In the given exercise, we use a specific function to calculate the gravitational force between the Moon and an object located at a certain distance above its surface. The formula provided, \[ F(x) = \frac{350}{x^2} \]embodies this gravitational force as a function of distance. Here, \( F(x) \) represents the gravitational attraction in newtons, and \( x \) indicates the distance in millions of meters. This formula tells us that the gravitational force is inversely proportional to the square of the distance, highlighting how gravity weakens as objects move further apart.

Inverse Square Law

The inverse square law is a key principle in physics, particularly with forces that operate over a distance, like gravity and light. The law states that a specified physical quantity or intensity is inversely proportional to the square of the distance from the source of that physical quantity. In simpler terms, as you double the distance from the source, the strength of the effect becomes one-fourth.
In the exercise, the formula \[ F(x) = \frac{350}{x^2} \]serves as an excellent example of this law in action. It clearly shows that the gravitational force \( F \) decreases with the square of the increase in distance \( x \). This means if you increase the distance between the objects by a factor of two, the force decreases by a factor of four, highlighting how quickly gravitational force diminishes with increasing separation.

Graphing Functions

Graphing functions can help us visually understand the behavior of relationships between variables. In the case of the given gravitational function \[ F(x) = \frac{350}{x^2} \],the graph shows how the force \( F(x) \) changes with distance \( x \).
By plotting key points, such as

• \( x = 1 \) where \( F(1) = 350 \)
• \( x = 2 \) where \( F(2) = 87.5 \)
• \( x = 10 \) where \( F(10) = 3.5 \)

we capture the curve's shape clearly. The curve starts at a high point and rapidly decreases, flattening out as \( x \) grows larger. Practically, this shows the force is quite significant when distances are small but diminishes swiftly as the distance increases, visualized by the steep initial drop in the curve.

Asymptotic Behavior

Asymptotic behavior refers to how a function behaves as it approaches a particular line called an asymptote. In this context, it describes what happens to the function \( F(x) = \frac{350}{x^2} \) as \( x \) becomes very large or approaches zero.
When \( x \) approaches zero, the value of \( F(x) \) grows infinitely large, representing a vertical asymptote. However, practically, this scenario is impossible since an object cannot be zero distance from the Moon's surface.
As \( x \) increases towards infinity, the gravitational force \( F(x) \) approaches zero. This is a horizontal asymptote, illustrating that while the gravitational pull never entirely vanishes, it diminishes to an almost negligible level as distance grows. Understanding this asymptotic behavior helps us interpret the graph and the physical implications of the gravitational force over large distances.