Problem 43 - Calculus Early Transcendentals: Pearson New International Edition

Question

The path of a certain comet is a parabola with the sun at the focus. The angle between the axis of the parabola and a ray from the sun to the comet is \(120^{\circ}\) (measured from the point of the perihelion to the sun to the comet) when the comet is 100 million miles from the sun. How close does the comet get to the sun?

Step-by-Step Solution

Answer

The comet gets as close as 50 million miles to the sun.

Step 1: Understanding the Geometry of the Problem

The problem involves a parabola with the sun at its focus. The angle formed by the axis of the parabola and a line connecting a point on the parabola to the focus is provided. Since it's a geometrical problem involving a parabola, we recall that the angle corresponds to the direction of the tangent line at the given point on the parabola.

Step 2: Identifying Key Parabola Properties

For a parabola, if the angle between the tangent and the line from the focus to a point is given, it relates to the eccentricity of the parabola. The angle defined in the problem forms an external angle with the vertex line, known as the supplementary angle of the focal angle, and for a parabolic trajectory, this focal angle is always 90°, leading to a triangle.

Step 3: Calculate Eccentricity Using the Angle

Using the supplementary angles principle, since the provided angle from the focus is 120°, internally, the minimum angle for the focal geometry is complementary to the supplementary, which is 60°. For parabolic orbits, this helps with the closest approach formula since we derive ratios using trigonometry for the parabolic section.

Step 4: Applying the Distance Formula for the Closest Approach

Using the distance formula where the cosine of the composed angles affects the spatial proximity, we compute: \( d = r \cos(\theta) \) where the minimum angle \(\theta\) is complementary, i.e., 60°. Here, \(r = 100\) million miles: \(d = 100 \, \text{million miles} \times \cos(60^{\circ}) = 100 \, \text{million miles} \times 0.5 = 50\text{ million miles}\).

Step 5: Conclusion

From our computation, the distance we calculated represents the perihelion distance, which is the minimum distance between the comet and the sun. Therefore, we determine that the comet gets as close as 50 million miles at its closest approach.

Key Concepts

Parabolic Trajectories

Parabolic trajectories are fascinating arcs found in nature and mathematics. An object follows a parabolic path when it is under the influence of a force like gravity. In the case of a comet orbiting the sun, this path is determined by the gravitational pull. The sun acts as a focal point, and the comet's path traces out a parabola.

• The trajectory's shape depends on the gravitational influence and the initial velocity of the comet.
• A parabolic path is a type of conic section, which also includes ellipses, circles, and hyperbolas.

In our exercise, we encounter a parabolic trajectory as the comet moves around the sun. This illustrates how celestial bodies, like comets, can follow predictable paths due to gravitational forces. Understanding these trajectories helps scientists predict comet behavior and anticipate their position at any given time.

Eccentricity of Parabolas

Eccentricity is a measure of how "stretched" a conic section is. For a parabola, the eccentricity is always equal to 1. This means no matter how you look at a parabola, its shape is consistent with its defining property of having a focus and a directrix.

• Eccentricity of 1 signifies the unique nature of parabolas among conic sections.
• Unlike ellipses (eccentricity less than 1) and hyperbolas (eccentricity greater than 1), parabolas remain constant in eccentricity.

Understanding eccentricity helps grasp why a comet's path around the sun takes on a parabolic shape. The problem provided uses the relationship between angles and parabolic properties to describe the comet's orbit. This approach demonstrates how mathematical properties of conic sections apply to real-world celestial movements.

Perihelion Distance

Perihelion distance is the closest point in the orbit of a celestial body to the sun. For our comet, this was calculated using geometric relationships and trigonometric principles. The exercise used the angle related to the parabolic shape and the distance of 100 million miles from the comet to the sun.

• Understanding angles: The angle between the axis and the line from the focus helps calculate how close the path gets to the focal point, which is the sun in this context.
• Using trigonometry: Calculating distances involves using the cosine of angles to determine the perihelion distance.

The calculated 50 million miles as the perihelion distance emphasizes the importance of orbits in astronomy. It shows how the closest approach of celestial bodies is predictable, which is vital for forecasting events in our solar system.