Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

Physics Independent Project  

Hello Students,  This resource packet includes a project that you can work on independently at home. You should also have project packets for some of the other courses you are enrolled in. These projects are standards-aligned and designed to meet the Remote Learning instructional minutes guidelines by grade band.   

High School Physics: What is the best Earth and Mars orientation to launch a spaceship to get to Mars? Estimated Time  ~225 minutes 

Grade Level Standard(s) 

HS PS2-4 Use mathematical representations of Newton’s Law of Gravitation and Coulomb’s Law to describe and predict the gravitational and electrostatic forces between objects.  HS ESS1-4 Use mathematical or computational representations to predict the motion of orbiting objects in the solar system 

Caregiver Support Option 

Caregivers can support by providing push pins, string, cardboard, and a ruler. They can also assist with drawing of circles and with the mathematical calculations. 

Materials Needed  Paper and Pencil, Calculator, ~30 cm Length of String, Push Pins, Cardboard, Ruler 

Question to Explore  What is the best Earth and Mars orientation to launch a spaceship to get to Mars? 

Student Directions  Directions are given throughout the packet. Please record your responses on a separate sheet of paper unless otherwise indicated. 

 Introduction Reading Adapted From: The Mars 2020 rover has a new name: Perseverance Devin Coldewey@techcrunch, March 5, 2020  Investigation 1 Introduction Adapted From: Kepler's Laws and Newton's Laws, Mount Holyoke College, AST 223/23 Planetary Science Course Page, by Darby Dyar.  Investigation 2 Activity Adapted From:  Let's Go to Mars! Calculating Launch Windows  Education.com - Orbital Eccentricity | Science Project Science Friday: Scale Solar System Orbits—And Satellites!     

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ Introduction  The next NASA rover to go to Mars has shed its code name and assumed a new one, sourced from the ingenuous youth of our nation. Keeping with the tradition of using virtues as names, the Mars 2020 rover will henceforth be known as “Perseverance.”  This particular virtue was suggested by Alexander Mather, a middle-schooler in Virginia. He and some 28,000 other kids proposed names in an essay contest last year. The final nine contenders were: Endurance, Tenacity, Promise, Vision, Clarity, Ingenuity, Fortitude, Courage and, of course the winner, Perseverance.  “Like every exploration mission before, our rover is going to face challenges, and it’s going to make amazing discoveries. It’s already surmounted many obstacles to get us to the point where we are today,” said Thomas Zurbuchen, NASA’s associate administrator of the Science Mission Directorate, in a news release. “Alex and his classmates are the Artemis Generation, and they’re going to be taking the next steps into space that lead to Mars. That inspiring work will always require perseverance.” 

 NASA scientists working on this mission have not only had to design and name the rover, but they have also needed to figure out the exact date in the near future that will be best to launch the spacecraft that will carry the Perseverance rover to Mars. When planning a launch to Mars, it is critical to take into account the relative positions of Earth (inner planet) and Mars (outer planet) at the time of launch. In addition, sending a mission to Mars is an incredibly expensive task that requires enormous amounts of fuel, so NASA pays close attention to choosing the best route to conserve fuel cost. As a result, to help the 

NASA scientists, your task in this packet is to figure out the answer to the driving question, “What is the best Earth and Mars orientation to launch a spaceship to get to Mars?”  

1. The five diagrams below show five different orientations of Earth and Mars that the NASA team is debating on as the best time point to launch a spacecraft to Mars. Here is some key information about the system: 

● Earth’s orbital radius is smaller than Mars’s, which means in the diagrams below, Earth is on the inner orbit and Mars is on the outer orbit. 

● Earth and Mars orbit the Sun counterclockwise (in this diagram), as indicated by the arrows on each planet. 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

Option A  Option B  Option C  Option D Option E

     

 2. On each of the five diagrams above, draw the path that you think a spaceship would take to 

travel from Earth to Mars. Be sure to indicate where you think Mars would be when the rocket arrives there. Record your response on a seperate sheet of paper. 

 3. Which of the five options do you think would be the best orientation of Earth and Mars to 

launch the spaceship? Why? Record your response on a seperate sheet of paper.  Investigation 1: What role does gravity play in picking the best launch orientation?  To help us figure out the best orientation of Earth and Mars to launch a spaceship to Mars, it will first help us to figure out the different ways that Earth and Mars orbit the Sun.  The Earth and Mars have been orbiting the Sun since the beginning of the solar system about 4.5 billion years ago. But why do they stay in these defined orbits and why don’t they just fly off into space? The reasons are complicated but the first good explanation was provided by one of the greatest scientists ever, Isaac Newton, who lived in England about 300 years ago. Newton realized that the reason the planets orbit the Sun is related to why objects fall to Earth when we drop them. The Sun's gravity pulls on the planets, just as Earth's gravity pulls down anything that is not held up by some other force and keeps you and me on the ground. Newton’s discovery that the force of gravity that acts on an object on Earth as well as the planets orbiting the Sun led to him to formulate the Universal Law of Gravitation, which is usually stated as an equation:  

F g = G r2M M1 2  

 where Fg is the attractive gravitational force between two objects of mass M1 and M2 separated by a distance r. The constant G in the equation is called the Universal Constant of Gravitation. The value of G is 6.67 X 10-11 meters3 kilograms-1 seconds-2.  

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ Interestingly, Newton’s Law of Gravitation can also be used to figure out the velocity of a planet in its orbit around the Sun, if you assume the orbit of a planet is circular. When you combine Newton's Law of Gravitation and circular acceleration, you can derive the relationship between velocity of a planet, its mass, and the radius of its orbit. You get the equation shown below:  

v = √ rGm1  

 4. To help us figure out the relative motion of the Earth and Mars, complete the table below to 

find the force of gravitational attraction between the Sun and each planet as well as the orbital velocity of each planet. Complete your work on a seperate sheet of paper. 

 

Planet  Planet Mass (kg) 

Mass of the Sun  (kg) 

Orbit Radius  (m) 

Fg (N)   F g = G r2M M1 2  

Orbital Velocity (m/s) 

v = √ rGm1  

Venus  4.87 x 1024  1.989 × 1030  1.082 x 1011  5.522 × 1022  3.5×104  

Earth  5.97 x 1024  1.989 × 1030  1.496 x 1011     

Mars  0.642 x 1024  1.989 × 1030  2.279 x 1011     

Jupiter  1898 x 1024  1.989 × 1030  7.786 x 1011     

Saturn  586 x 1024  1.989 × 1030  14.34 x 1011     

 5. What patterns or relationships do you see in the data? Consider discussing: 

a. How the orbital velocity relates to the orbital radius. b. How the gravitational force relates to the orbital radius. 

6. Using the data above, predict the approximate relative positions of Earth and Mars after 3 months (¼ of an Earth orbit) and 6 months (½ of an Earth orbit). 

Hypothetical Starting Orientations 

Orientation at 3 Months  Orientation at 6 Months 

 

   

 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 7. Explain your two predictions on a separate sheet of paper. Cite evidence from the data table 

in Question 4 to support your predictions.  The spaceship we are launching will use its thrusters to escape Earth’s gravitational pull, and then to conserve fuel, instead of using its fuel and thrusters to navigate the trip to Mars, it will enter into an orbit around the Sun.   

8. In the space below, sketch with a dashed line what you think the orbit of the spacecraft will look like. Explain what you drew in the space to the right, and be sure to use Newton’s Law of Gravitation to explain why you think using an orbit is more-fuel efficient for the spaceship than using fuel and thrusters. 

 

Diagram of Spaceship Orbit 

Why Newton's Law of Gravitation can explain why it is more fuel efficient to enter into an orbit than it is to use thrusters. 

 

 

 9. Looking across everything in Investigation 1, what new ideas do you have that will help inform 

your choice of the best orientation of Earth and Mars for the spaceship launch? Record your response on a separate sheet of paper. 

 Investigation 2: What would an elliptically-shaped spaceship pathway look like?  Describing Planetary Orbits With Kepler’s First Law  Although we have chosen to model planetary orbits as circles so far in this packet, in reality, all planetary orbits are ellipses. These elliptical paths are described by Kepler’s First Law, which simply states that planets move in ellipses.   

10. To model the elliptical orbits of planets, you will use string and push pins to draw ellipses and see how their shapes change depending on several parameters. Follow the directions below: 

 ● Using a piece of the provided graph paper, draw a straight line across the paper long 

ways, through the center of the paper. ● Mark two dots on the line 10 cm apart at the center of the paper. 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ ● Place the paper on top of a piece of cardboard about the same size as the paper and 

stick the pushpins into the dots. Don’t worry if your cardboard isn’t a perfect size because it is only there to hold the pushpins. 

● Cut a string to a length of 25 cm. Tie the loose ends of the string together to make a loop. Be sure not to tie off too much extra string so that your loop stays as big as possible. 

● Drape the string loop around the pushpins. Place the pencil inside the loop as well and use it to stretch the loop into a triangle. 

● Keeping the string taut, guide the pencil around the pushpins while tracing out a path on the paper below. Continue around the pins until you’ve drawn a closed loop.  

 11. Describe the shape of the curve you’ve drawn. Is the diameter across the loop always the 

same, or does it change? Where is it widest? Where is it narrowest? Record your response on a seperate sheet of paper. 

 ● Repeat the above steps on the same graph paper with the pins at different distances 

from each other: 0 cm (just use one pin), 33 cm, and 12.7 cm. Trace each of these lines with a different colored pencil or pen, if you have one.  

 12. How does the shape of the loop change? How does the distance across the widest and 

narrowest parts change as the pins get closer together? Record your response on a separate sheet of paper. 

 You have four ellipses on the poster board. Each pushpin marks a point in the ellipse called a focus (plural: foci). The widest diameter across the ellipse is called the major axis; the narrowest diameter is known as the minor axis. The foci sit along the major axis, equidistant from the center of the ellipse. As the foci get closer together, the ellipse looks more like a circle. An ellipse with only one focus is a circle (the major and minor axes are the same length). 

   In 1609, Johannes Kepler figured out that the planets travel along elliptical paths with the sun sitting at one focus of the ellipse. He called this his First Law of Planetary Motion. As a planet moves along its orbit, the distance between it and the sun changes. The point on the ellipse where the planet is closest to the sun is called the perihelion; the point where it is farthest is the aphelion. The Earth passes its perihelion in early January and goes through aphelion in early July. On your ellipses, can you mark where the perihelion and aphelion might be? 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________  The eccentricity of an orbit is a single number, between 0 and 1, which describes how stretched out the orbit is. Zero means the orbit is perfectly circular. An eccentricity close to 1 means the orbit is extremely elongated; only comets coming from the outer reaches of the solar system get close to this value.  You can calculate the eccentricity of your ellipses using the following equation:  

e = a + pa − p

 

● where e is the eccentricity, ● a is the aphelion distance, and ● p is the perihelion distance. 

 13. For each ellipse, pick a focus where the Sun should sit (either one will do) and draw in a Sun. 

Then, measure the aphelion and perihelion distances with your ruler. Calculate the eccentricity, and record your results in Table 1. 

 

Ellipse  Distance Between Foci (cm) 

Aphelion  Perihelion  Eccentricity 

A  0       

B  7.6       

C  10.2       

D  12.7       

 14. How does the eccentricity change as the foci get farther apart? Record your response on a 

separate sheet of paper. 15. Now you know enough to figure out the eccentricities of the planets in the solar system. Table 

2 lists the aphelion and perihelion distance of all the major planets. The distances are in astronomical units (AU), where 1 AU is the average distance between the Earth and Sun (93 million miles). 

 

Name  Aphelion (AU)  Perihelion (AU)  Eccentricity 

Mercury  0.47  0.31   

Venus  0.73  0.72   

Earth  1.02  0.98   

Mars  1.67  1.38   

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

Jupiter  5.46  4.95   

Saturn  10.12  9.05   

Uranus  10.08  18.38   

Neptune  30.44  29.77   

 Describing the Spaceship Journey to Mars Using Kepler’s First Law To help us figure out what pathway of the spaceship would take, it can be helpful to think of sending a spacecraft to Mars like throwing a pass on the football field. Once the thrower releases the ball, both the ball and the receiver are in motion. The thrower cannot guide the ball once it is in flight, so they have to anticipate where the receiver will be in the future and throw the ball to arrive at that location.  The same idea applies to sending a spacecraft to Mars. Once the spacecraft leaves Earth’s orbit, we want it to travel to Mars using only the gravitational force from the Sun. Taking a different pathway would require the spaceship to work against gravity, which would require using more fuel. To best take advantage of a gravitational orbit for the spaceship, we must time the launch so that Mars and the spacecraft both arrive in the same location at the same time, and Mars can then “catch” the spacecraft. This special path is called a Hohmann Transfer Orbit, which was determined by Walter Hohmann, an architect whose hobby was celestial mechanics.   Hohmann demonstrated mathematically that in many cases, the most efficient flight path between two different orbits was along an elliptical path from one side of the Sun to the other, beginning and ending at each orbit’s furthest distance from the sun. This elliptical spaceflight maneuver uses the angular velocity of the earth’s orbit as well as the sun’s own gravity to “throw” a satellite into an elliptical orbit that intersects the orbit of the target planet.  In order to model the Hohmann Transfer orbit, we will once again simplify the models of the orbits of Earth and Mars. Even though we determined that their orbits have small eccentricity values according to Kepler’s First Law, we will now model their orbits as circular to simplify the calculations for determining the Hohmann Transfer Orbit.  

16. To start our model of the Hohmann Transfer Orbit, we need to first model the orbits of Earth and Mars. On a new piece of graph paper, use your push pins to now create the circular orbits of Earth and Mars according to the directions in the table below. Fill in the right column with the values you determine as well. 

  

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

Draw Earth’s Orbit Use a pushpin, loop of string, and a pencil to draw a circle representing Earth’s orbit, with the center representing the sun. Next, use a ruler to draw a straight line from the center to the outside of the circle (the Earth’s orbital radius). 

 

Why do you think it is called the orbital radius?  

Label Your Drawing  Label the sun, Earth’s orbit, and Earth’s orbital radius in your diagram. Use a ruler to measure Earth’s orbital radius in centimeters, which in real life is equivalent to 1 AU. 

 

Measure your drawing:  Earth’s orbit in cm ______  Earth’s orbit in AU ______  

Find The Scale Of Your Drawing The scale of your drawing is a ratio that indicates precisely how small your drawing is compared to the real thing. In this case, the scale of your drawing is a ratio of the number of centimeters (cm) that represent 1 AU. 

Example If the orbital radius you drew for Earth measures 3cm:  1AU = 3cm  Scale = 3cm / 1AU  Drawing scale = 3cm/AU  

Calculate scale:  1AU = _________cm  Scale = _______cm/ 1AU  Scale = ______ cm/AU  

Find the Orbit of Mars For Your Drawing Mars also orbits the Sun at a radius of 1.52 AU. Multiply the orbital radius length of Mars (in AU) by the scale of your drawing (cm/AU) to figure out how big to draw the orbit of Mars. 

Mars Orbital Radius: __________AU   Drawing scale: _________cm/AU   Calculate Mars’s scale orbital radius:   Mars Scale Radius = (scale) * (orbital radius) = _______  

 17. With the orbits of Earth and Mars established, you will now draw the Hohmann Transfer Ellipse. 

Follow the directions in the table below to add the transfer to your diagram on the same piece of graph paper, and fill in the values in the right-hand column of the table. 

 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

Draw The Major Axis Of Your Hohmann Ellipse Use a ruler to draw a horizontal line through your solar system illustration, from the far side of one orbit through the sun, to the opposite side of the other orbit. This line spans what will be the widest part of your finished Hohmann ellipse, called the “major axis.”  

 

Measure And Label The Features You Need In Order To Draw Your Hohmann Ellipse  ● Measure the length of the major axis (the 

orbital radius of Earth + the orbital radius of Mars) in AU. 

● Calculate the center of the major axis. Divide the length of the major axis by two and place an “X” at the middle of the major axis. 

● Measure the focal distance (distance from the Sun to the center point of the major axis). The Sun is one focus of the Hohmann ellipse. 

● Place an “O” at the position of the second focus of the Hohmann ellipse. This focus is located on the opposite side of the center point from the Sun, one focal distance from the center point. 

 Draw The Hohmann Ellipse Measure a loop of string that when taped or tied reaches from one focus, past the center, to the opposite side of the Hohmann ellipse.  Place a pin at the position of the two foci (the Sun and the second focus), place your loop over both pins, then draw the ellipse. When you are done, erase the second half of your Hohmann transfer ellipse—a Hohmann transfer only goes halfway around the sun. 

  Convert all of the distances in your Hohmann transfer illustration to AU by dividing each distance (in cm) by the scale factor (in cm/AU) of your illustration that you calculated.   Drawing scale: ___________ cm/AU   Length of major axis at scale: __________ cm   Length of major axis in reality : __________ AU  (= length of major axis / scale)   Focal distance: _________ cm   Focal distance: _________ AU 

    

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ Investigation 3: How long would it take to complete a journey to Mars, and when should the journey start? Next, you will use Kepler’s Third Law, the Law of Harmony, to determine the period of the Hohmann transfer orbit and then the travel time to Mars along this orbit.   Kepler’s Third Law states that the square of the period of any planet is proportional to the cube of the semi-major axis of its orbit, P2 ∝ a3. More specifically, Kepler's Third Law is written as P2 = ka3 with k being the constant of proportionality. Using Earth as an example, we can measure P in years and a in astronomical units (AUs) so P = 1 year and a = 1 AU. Thus for calculations involving Earth with k = 1, P2 = ka3 can simply be expressed as:  

P2 = a3  

18. An example of using Kepler’s Third Law to calculate the Hohman transfer time from Earth to Jupiter is shown on the left in the table below. Jupiter is further from the Sun than Mars, so the predictions using these calculations will be much different than those for Mars. However, the same process will apply for both. Use this example to calculate the period of the Hohmann transfer from Earth to Mars on the right side of the table. 

Jupiter  Mars 

Jupiter’s orbital radius is 5.2 AU. To calculate the semi-major axis a of Jupiter’s orbit:  

2a = rEarth + rJupiter  2a = 1 AU + 5.2 AU 

2a = 6.2 AU a = 3.1 AU 

Semi-major axis of Mars’s orbit: 

To determine the period (one full orbit) of the Hohmann Transfer Orbit, use Kepler’s Third Law:  

P2 = a3 P2 = (3.1 AU)3 

P = ~5.46 years P = ~1,993 days 

Period of the Hohmann Transfer orbit for Mars: 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

The full period of this Hohmann transfer orbit is 1,993 days. As you saw in your diagram of the Hohmann Transfer Orbit, the transfer itself requires only half of one orbit. So travel to Jupiter on the Hohmann orbit would take ½ of the period of the Hohmann Transfer Orbit.  

P = ~1,993 days ½ P = ~997 days 

Travel Time to Mars on the Hohmann Transfer Orbit: 

 19. Next, we must determine the correct relative position of Earth and Mars for launch. We will 

begin again with Jupiter as an example.  

Jupiter  Mars 

Jupiter completes one revolution around the Sun (360°) in 4331 days. 

Mars completes one revolution around the Sun in 687 days. 

To calculate the distance in degrees that Jupiter travels every Jupiter-day:  

360°/4331 days = 0.0831° per day 

Distance in degrees Mars travels per Mars-day:    

We know that the Hohmann transfer from Earth to Jupiter is 997 days.   

997 days X 0.0831° per day = 82.9°  Jupiter will travel ~83° during the time the spacecraft travels from Earth to Mars.  

Calculate how many degrees Mars will travel during the time of the Hohmann transfer. 

The diagram below shows the path of the Hohmann transfer orbit. The spaceship should arrive at Jupiter when Jupiter is at position 180° relative to Earth’s initial position of 0°.  

What should Mars’ position be relative to Earth at launch? 

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ 

  

If the spaceship arrives at Jupiter at position 180°, we can determine the relative angle Jupiter should be to Earth when the spaceship is launched by subtracting the distance traveled by Jupiter (in degrees) from 180°.    

180° - 83° = 97°  We can conclude that the ideal time to launch a spacecraft to Jupiter using the Hohmann transfer is when Jupiter is 97 degrees ahead of Earth in its orbit. 

The diagram below shows Jupiter’s position relative to Earth’s at launch for the Hohmann 

transfer. 

 

Draw a diagram of where Mars should be relative to Earth at launch.   

  

    

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ Creating Final Evidence-Based Arguments  

20. Return to the five options of starting orientations of Earth and Mars from the beginning of this task. They are once again shown below.

Option A  Option B  Option C  Option D Option E

     

Complete the following on a separate sheet of paper: 

Claim: Record which of the above options you think is best supported by the evidence. Evidence: Record at least three pieces of specific and appropriate evidence from the data sets analyzed. Reasoning: Record how and why you think this option is most supported, including mathematical reasoning that backs up your ideas. Rebuttal: Record one alternative option and provide one piece of evidence and reasoning that refutes this option. Final Reflection Questions  

21. What new ideas do you have about the factors NASA scientists consider when they plan a journey to a new planet? 

22. Most models or illustrations are simplified representations of reality, which can be helpful for simplifying information and conveying a single thing clearly. The things that are left out of the models can lead to the model having limitations. What limitations do the models in this packet have? Try to come up with at least five types of information that your orbit illustrations do not show and explain why the simplifications could be important. 

23. The Hohmann transfer is very useful for moving between two different orbits around the sun (called “heliocentric orbits”), but it can also be used to transfer between earth orbits that satellites use, such as low-earth orbit for the International Space Station. Research the orbital radius of the International Space Station, then plan a Hohmann transfer for a rocket that needs to dock with the space station to deliver supplies. 

    

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________ Graph Paper for Investigation 2  

      

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Student Name: _______________________ School Name:_____________________ Teacher Name:_____________  

  

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