1. Module 5 Project

Mathematics 20-1 Module 5

Module 5: Radicals

 

Module 5 Project: Plan a Planet

 

This is an image of a green planet.

Digital Vision/Thinkstock

 

Introduction

 

Scientists have accumulated evidence of several hundred planets circling stars outside Earth’s solar system. These planets are called exoplanets. Most are huge and are so close to their star that they would be too hot to support life. Because huge planets are easier to detect, exoplanets were the first to be discovered.

 

Scientists have found evidence for a planet that appears to be cool enough to support life. It is called Gliese 581c. Gliese 581c orbits a sun that is cooler than Earth’s sun, is about 11 light-years away, and is in the constellation Libra.

 

In this project you will use radical equations to find out more about Gliese 581c. Then you will use the radical equations to design a planet that could support life as the planet travels around a distant star.

 

Activity 1
  1. The mass of a planet is given by the formula , where D is the average density of the planet and r is the radius in metres. Rearrange this equation so that r is expressed in terms of D and m.

  2. Gliese 581c is about 5.4 times the mass of Earth. If Gliese 581c has a density similar to solid rock, which is 5500 kg/m3, and the mass of Earth is 6 × 1024 kg, what is the radius of Gliese 581c?

  3. Calculate the radius of the planet if the density is similar to that of the ice world, Neptune, with D = 1600 kg/m3.

  4. The acceleration due to gravity on Earth is about 9.8 N/kg. The acceleration due to gravity on a planet can be calculated from the formula , where m1 is the mass of the planet and r is the radius. G is the universal gravitational constant 6.67 × 10−11 N•m2/kg2. Rearrange the formula so that r is expressed in terms of g and m, and then calculate what the radius of Gliese 581c would have to be for the acceleration of gravity to be the same as on Earth.
Activity 2

 

This is an image of a make-believe planet and sun.

Hemera/Thinkstock

 

Now you get to design your own planet. Assume your planet is rotating around a star with a radius of 800 000 km and a temperature of 6000 K, slightly larger and hotter than Earth’s sun. The planet should have liquid water to support life, so it must have an average temperature somewhere between 0°C (273 K) and 100°C (373 K).

 

The average temperature on the planet in degrees Kelvin can be calculated by the formula , where d is the distance of the planet from the star in kilometres, and R is the radius of the star in kilometres. The temperature of the star, Ts, is also in degrees Kelvin.

  1. Decide the average temperature, in degrees Kelvin, you want for your planet. Rearrange the temperature formula to isolate d, and then calculate how far from the star your planet will have to be.

  2. Decide the density of the planet and record your decision. Earth’s density is somewhat higher than the density of solid rock at 5500 kg/m3 because of the iron core. Your planet probably should be too. The iron core helps maintain Earth’s magnetic field, which shields Earth’s population from the bombardment of most cosmic rays.

  3. Decide a mass for your planet and record your decision. If the mass is too small, the gravity cannot hold gas molecules on the planet and there will be no atmosphere. If the planet is too large, the gravitational force would hamper the movement of terrestrial life.

  4. Using a formula you already used in Module 5 Project: Plan a Planet, calculate the radius of your planet.

  5. Using a formula you already used in Module 5 Project: Plan a Planet, calculate the acceleration due to gravity on your planet.

  6. Calculate the maximum distance the planet could be from the star and still have liquid water.

  7. Describe the features of your planet, including topography, vegetation, other life forms, and climate. Use diagrams and art if needed to communicate your concepts.
Conclusion

 

Write a brief conclusion to your project. Include your personal reaction to doing the project. Explain how you felt about your project during and after completion and why you felt this way.

 

Assessment

 

Your project will be evaluated by your teacher using the following rubric. Read the rubric carefully and ensure that you complete all requirements to the best of your ability.

 

RUBRIC FOR MODULE 5 PROJECT: PLAN A PLANET
Score Operations on Radicals and Radical Expressions Completion Communication: Final Presentation Written Explanations

3

Meets the Standard

 

 All required operations on radicals and radical expressions are completed and correct. All aspects of the project are completed. The work is presented in a neat, clear, organized fashion that is easy to read and/or see. Explanations are detailed and clear.

2

Approaches the Standard

 

 There may be some serious math and/or calculation errors or flaws in reasoning. All but one aspect of the project are completed. The work is presented in an organized fashion but may be hard to read and/or see at times. Explanations are clear.

1

Below the Acceptable Standard

 

 There are major math and/or calculation errors or serious flaws in reasoning. All but two aspects of the project are completed. The work appears sloppy and/or disorganized. It is hard to know what information goes together. Explanations are a little difficult to understand but include critical components.

INC

Does Not Meet the Minimum Standard		 There is no understandable attempt at using mathematical representations. Several aspects of the project are not completed. There is no understandable presentation of project work.

Explanations are difficult to understand and are missing several components or were not included.

Total Score

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/12

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