1.12.2: Explorations

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Mar 8, 2021
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- 1.12.1: Illustrations
- 1.12.3: Problems

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Exploration 1: Different $x_{0}$ or $v_{0}$ for Planetary Orbits

This Exploration shows $10$ identical planets orbiting a star. The initial position of the planets can be set at $t = 0$ time units when the planets are on the $x$ axis. The difference in orbital trajectory, therefore, is due to the planets' initial velocities (in this animation GM $= 1000$ ). Restart.

As you vary the initial positions of the planets, how do the orbital trajectories change?
Find a planet with circular motion. What is the period for this motion?
What happens to the orbit when $x$ gets really small?
What happens to the orbit when $x$ gets really large?

This part of the Exploration shows $10$ identical planets orbiting a star. The initial velocity of the planets can be set at $t = 0$ time units when the planets are on the $x$ axis.

As you vary the initial velocities of the planets, how do the orbital trajectories change?
Find a planet with circular motion. What is the period for this motion?
What happens to the orbit when $v$ gets really small?
What happens to the orbit when $v$ gets really large?

Exploration authored by Mario Belloni and modified by Emmy Belloni.

Exploration 2: Set Both $x_{0}$ and $v_{0}$ for Planetary Orbits

This Exploration shows a planet orbiting a star. The initial position in the $x$ direction and the initial velocity in the $y$ direction of the planet can be set at $t = 0$ time units when the planet is on the $x$ axis. The difference in orbital trajectory, therefore, is due to the planet's initial position and velocity (in this animation GM $= 1000$ ). Restart.

As you vary the initial velocity of the planets, how do the orbital trajectories change?
What happens to the orbit when $x_{0}$ gets really small (keep $v_{0y} = 10$ )?
What happens to the orbit when $x_{0}$ gets really large (keep $v_{0y} = 10$ )?
What happens to the orbit when v_0y gets really small (keep $x_{0} = 5$ )?
What happens to the orbit when v_0y gets really large (keep $x_{0} = 5$ )?
Find the condition for circular motion.
For circular motion, what is the period?
During each of your investigations, what was happening to the angular momentum as time passed? Why?
Make $x_{0} = 10$ . Then for small $v_{0}$ , what type of orbit occurs?
For $x_{0} = 10$ , what $v_{0}$ makes the orbit circular?
As you increase $v_{0}$ ( $x_{0} = 10$ ), the orbit changes shape. What shape does it have just beyond the speed required for circular orbit?
As you increase $v_{0}$ ( $x_{0} = 10$ ) even further, you eventually reach a condition of "escape." Use energy considerations to predict what this escape velocity should be.
For any circular orbit, predict (and then check on the graphs) how the magnitude of potential energy compares to kinetic. Likewise for escape velocity.
For a given orbit, you should note that the angular momentum remains constant. How does this relate to the other quantities in the table (under the simulation window), and discuss what is meant by the angle "theta."

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Exploration 3: Properties of Elliptical Orbits

A planet (green) orbits a star (yellow) as shown in the animation. Restart.

On a piece of paper sketch vectors for the velocity, radial component of acceleration, and tangential component of acceleration for the planet. The length of the vectors should be indicative of the magnitude of the vectors.

Rank the points, A–E, according to the speed of the planet at that point.
Rank the points, A–E, according to the gravitational potential energy of the planet.
Rank the points, A–E, according to the kinetic energy of the planet.
Rank the points, A–E, according to the total energy of the planet.
At which of the points, A–E, is the planet's acceleration in the same direction as the velocity?
What can you say about the direction of the planet's acceleration at any point on its path? Would you call this acceleration a tangential acceleration or a radial acceleration?

Click here to view the velocity vector (blue) and acceleration vector (red). Compare what you see to your answers (a)–(f).

Exploration authored by Aaron Titus with support by the National Science Foundation under Grant No. DUE-9952323 and placed in the public domain.

Exploration 4: Angular Momentum and Energy

A planet (with a mass equal to that of Earth) orbits a star as shown in the animation (position is given in astronomical units [A.U.] and time is given in years). Along with the animation is a graphical depiction of the energy of the planet. Three curves are shown: in black, the total effective potential energy; in blue, the gravitational potential energy; and in red, the effective rotational potential energy represented by the term: $L^{2}/2mR^{2}$ . The teal line represents the total energy of the planet as a function of distance to the central star, $R$ . Restart.

What happens to the red curve as the initial speed of the planet is changed?
What happens to the blue curve (the gravitational potential energy) as the initial speed of the planet is changed?
What happens to the teal curve (the total energy) as the initial speed of the planet is changed?

Now consider the total energy and angular momentum calculated in the table. Look at the circular, bound, and unbound cases.

How do the values for total energy and angular momentum change when the type of orbit is changed?
Can you find a general rule for whether an orbit is bound?
Feel free to explore different values of the initial velocity.

When you get a good-looking graph, right-click on it to clone the graph and resize it for a better view.

Physlets were developed at Davidson College and converted from Java to JavaScript using the SwingJS system developed at St. Olaf College.

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Exploration 1: Different x0x_{0} or v0v_{0} for Planetary Orbits

Exploration 2: Set Both x0x_{0} and v0v_{0} for Planetary Orbits

Exploration 3: Properties of Elliptical Orbits

Exploration 4: Angular Momentum and Energy

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Exploration 1: Different $x_{0}$ or $v_{0}$ for Planetary Orbits

Exploration 2: Set Both $x_{0}$ and $v_{0}$ for Planetary Orbits