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Chapter 5: Problem 5

A planet is revolving in an elliptical orbit around the Sun. Its closest distance from the Sun is \(r_{\min }\) and the farthest distance is \(r_{\max }\). If the velocity of the planet at the distance of the closest approach is \(v_{1}\) and that at the farthest distance from the Sun is \(v_{2}\), then \(v_{1} / v_{2}\) (1) \(\frac{r_{\max }}{r_{\min }}\) (2) \(\frac{r_{\min }}{r_{\max }}\) (3) \(\frac{r_{\min }+r_{\max }}{r_{\max }-r_{\min }}\) (4) none of these

Short Answer

Expert verified
(1) \(\frac{r_{\max}}{r_{\min}}\)

Step by step solution

01

Understand the Problem

The problem involves a planet orbiting in an elliptical path around the Sun. We're given the closest distance (periapsis, \(r_{\min}\)) and the farthest distance (apoapsis, \(r_{\max}\)) from the Sun, as well as the velocities at these points (\(v_1\) at \(r_{\min}\) and \(v_2\) at \(r_{\max}\)). We need to find the ratio \(v_1 / v_2\).
02

Apply Conservation of Angular Momentum

For elliptical orbits, the angular momentum \(L\) is conserved. Therefore, the angular momentum at the closest distance \(r_{\min}\) is equal to that at the farthest distance \(r_{\max}\). Mathematically, this is given by: \[ r_{\min} \times v_1 = r_{\max} \times v_2 \]
03

Solve for the Velocity Ratio

Using the conservation equation from Step 2: \[ v_1 = \frac{r_{\max} \times v_2}{r_{\min}} \] To find the ratio \(v_1 / v_2\), divide both sides by \(v_2\): \[ \frac{v_1}{v_2} = \frac{r_{\max}}{r_{\min}} \]
04

Compare with Given Options

The calculated ratio \(\frac{v_1}{v_2} = \frac{r_{\max}}{r_{\min}}\) matches option (1).

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Key Concepts

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

Conservation of Angular Momentum
In physics, the conservation of angular momentum is a crucial principle especially when considering orbital mechanics, such as a planet orbiting the Sun in an elliptical path. This principle states that if no external torque acts on a system, the total angular momentum remains constant.
To get a better grasp, let's consider a planet traveling in an elliptical orbit. The important takeaway here is that as the planet moves along its orbit, its distance from the Sun changes. However, the product of the radius (distance from the Sun) and the planet's velocity is constant. This means when the planet is closer to the Sun (at perihelion), it moves faster. And conversely, when it's farther from the Sun (at aphelion), it moves slower.
Mathematically, this is represented as:
  • For perihelion: \[ L = r_{\min} \times v_1 \]
  • For aphelion: \[ L = r_{\max} \times v_2 \]
Since angular momentum \( L \) is conserved, these equations allow us to establish relationships between the velocities and distances at these two points.
Kepler's Laws
Kepler's Laws of Planetary Motion are fundamental in the study of how planets move around the Sun. Let's simplify these laws:
1. **First Law** - The Law of Ellipses: Planets move in elliptical orbits with the Sun at one of the foci. This means orbits aren't perfect circles; they're slightly stretched, resembling an oval.
2. **Second Law** - The Law of Equal Areas: Often stated that a line segment joining a planet to the Sun sweeps out equal areas during equal intervals of time. This law explains why planets move faster when they are closer to the Sun and slower when further away.
3. **Third Law** - The Law of Harmonies: This law relates the time a planet takes to orbit the Sun with its average orbital radius. Precisely, it states the square of the orbital period is proportional to the cube of the semi-major axis of its orbit.
  • It's formalized as: \[ T^2 \propto a^3 \]
These laws allow us to make accurate predictions of the motion of planets and other celestial bodies in the solar system.
Planetary Motion
The motion of planets is a fascinating subject combining physics and astronomy. It involves the movement of celestial bodies, mostly under the influence of gravity.
Primarily, this motion is characterized by:
  • **Elliptical Orbits**: Planets do not follow a perfect circular path but rather an elliptical one, as Kepler explained.
  • **Gravitational Forces**: Isaac Newton further elaborated these motions with his Law of Universal Gravitation, stating every two objects in the universe attract each other with a force dependent on their mass and the distance between them.
  • **Orbital Velocity**: Defined as the speed a planet must travel to stay in a stable orbit. This changes as the planet moves through different sections of its orbit. Faster at closer distances and slower farther away due to conservation of angular momentum.
By understanding planetary motion, we gain insights into not just the mechanics of our solar system but also the universe at large, opening doors to explore extraterrestrial realms and phenomena.

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Most popular questions from this chapter

If due to air drag, the orbital radius of the earth decreases from \(R\) to \(R-\Delta R, \Delta R \ll R\), then the expression for increase in the orbital velocity \(\Delta v\) is (1) \(\frac{\Delta R}{2} \sqrt{\frac{G M}{R^{3}}}\) (2) \(-\frac{\Delta R}{2} \sqrt{\frac{G M}{R^{3}}}\) (3) \(\Delta R \sqrt{\frac{G M}{R^{3}}}\) (4) \(-\Delta R \sqrt{\frac{G M}{R^{3}}}\)

The value of acceleration due to gravity at the surface of the earth is \(9.8 \mathrm{~m} \mathrm{~s}^{-2}\) and its mean radius is about \(6.4 \times 10^{6} \mathrm{~m}\). Assuming that we could get more soil from somewhere, estimate how thick (in \(\mathrm{km}\) )would an added uniform outer layer on the earth have to have the value of acceleration due to gravity \(10 \mathrm{~m} \mathrm{~s}^{-2}\) exactly ? (Given the density of the earth's soil, \(\rho=5,520 \mathrm{~kg} \mathrm{~m}^{-3}\).)

Two bodies with masses \(M_{1}\) and \(M_{2}\) are initially at rest and a distance \(R\) apart. Then they move directly towards one another under the influence of their mutual gravitational attraction. What is the ratio of the distances travelled by \(M_{1}\) to the distance travelled by \(M_{2} ?\) (1) \(\frac{M_{1}}{M_{2}}\) (2) \(\frac{M_{2}}{M_{1}}\) (3) 1 (4) \(\frac{1}{2}\)

The gravitational force between two objects is proportional to \(1 / R\) (and not as \(1 / R^{2}\) ) where \(R\) is separation between them, then a particle in circular orbit under such a force would have its orbital speed \(v\) proportional to (1) \(\frac{1}{R^{2}}\) (2) \(R^{0}\) (3) \(R^{1}\) (4) \(\frac{1}{R}\)

A space station is set up in space at a distance equal to the earth's radius from the surface of the earth. Suppose a satellite can be launched from the space station. Let \(v_{1}\) and \(v_{2}\) be the escape velocities of the satellite on the earth's surface and space station, respectively. Then (1) \(v_{2}=v_{1}\) (2) \(v_{2}v_{1}\) (4) \((1),(2)\) and (3) are valid depending on the mass of satellite
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