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Chapter 4: Problem 6

Rank the following types of orbits in terms of the maximum speed of the orbiting object, from smallest to largest. a. elliptical, with semimajor axis \(R\) b. hyperbolic c. circular, with radius \(R\) d. parabolic

Short Answer

Expert verified
Circular, elliptical, parabolic, hyperbolic.

Step by step solution

01

Understand the types of orbits

Identify and understand the types of orbits provided: elliptical, hyperbolic, circular, and parabolic.
02

Recall the characteristics of circular orbit

In a circular orbit with radius \(R\), the speed is constant. It can be calculated using the formula \(v = \sqrt{\frac{GM}{R}}\).
03

Recall the characteristics of elliptical orbit

For an elliptical orbit with semimajor axis \(R\), the speed varies, being maximum at the closest point (periapsis) and minimum at the farthest point (apoapsis). The maximum speed is greater than that of a circular orbit with the same semimajor axis.
04

Recall the characteristics of parabolic orbit

For a parabolic orbit, the object is at escape speed. This speed is given by the formula \(v = \sqrt{2\frac{GM}{R}}\). Hence, it is greater than the speed in a circular orbit with the same radius.
05

Recall the characteristics of hyperbolic orbit

In a hyperbolic orbit, the object exceeds escape speed, making it faster than parabolic, elliptical, and circular orbits.
06

Rank the orbits

Based on the previous steps, rank the orbits from smallest to largest maximum speed: circular, elliptical, parabolic, hyperbolic.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

circular orbit speed
A circular orbit is when an object moves around another object in a perfect circle. The speed in a circular orbit with radius \(R\) is constant. We determine this speed using the formula:
\[ v = \sqrt{ \frac{GM}{R} } \]
Here:
  • \( v \) is the orbital speed.
  • \( G \) is the gravitational constant.
  • \( M \) is the mass of the object being orbited.
  • \( R \) is the radius of the orbit.
In simpler words, a smaller orbit radius results in higher speeds and vice versa. The speed is consistent throughout, making computation easier compared to other types.
elliptical orbit speed
An elliptical orbit resembles an elongated circle or an oval. Here, the speed of the orbiting object varies.
It moves fastest at its closest point to the object being orbited (periapsis) and slowest at its farthest point (apoapsis).
The maximum speed at periapsis is greater than the speed in a circular orbit with the same semimajor axis.
Calculating the exact speed at different points is complex, but knowing the object speeds up as it comes closer and slows down as it moves away is vital. This variability differentiates elliptical orbits from circular ones.
parabolic orbit speed
A parabolic orbit occurs when an object achieves escape velocity, meaning it can break free from the gravitational pull of the object it orbits.
The speed here is shown by:
\[ v = \sqrt{ 2\frac{GM}{R} } \]
This particular velocity is higher than that found in a circular orbit. Because
  • \( v \) is escape speed.
  • \( G \) is the gravitational constant.
  • \( M \) is the mass of the object being orbited.
  • \( R \) is the distance from the center of the object being orbited.
Understanding this allows us to see why a parabolic orbit's speed is crucial when considering escaping a planet's gravitational field.
hyperbolic orbit speed
In a hyperbolic orbit, an object travels at a speed exceeding escape velocity. This makes it faster than in parabolic, elliptical, and circular orbits. A hyperbolic orbit describes an object (e.g., a spaceship) that approaches, then leaves the orbited object, never to return.
Its speed can be generalized as beyond the escape speed, meaning
  • > \( \sqrt{ 2\frac{GM}{R} } \)
Since it surpasses the parabolic speed, which already is greater than that in circular and elliptical orbits, it ranks the fastest.

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Most popular questions from this chapter

As described in Math Tools \(4.4,\) tidal force is proportional to the masses of the two objects and is inversely proportional to the cube of the distance between them. Some astrologers claim that your destiny is determined by the "influence" of the planets that are rising above the horizon at the moment of your birth. Compare the tidal force of Jupiter (mass \(1.9 \times 10^{27} \mathrm{kg}\); distance \(7.8 \times 10^{11}\) meters with that of the doctor in attendance at your birth (mass \(80 \mathrm{kg}, \text { distance } 1 \text { meter })\)

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