# National Aeronautics and Space Administration

## Engineering Design

Problem 681:A Practical Application of Vector Dot and Cross Products Students work with coordinate vectors describing the corners of the roof of a house, calculate the area of the roof using dot products; calculate the normal vector to the roof using cross products; and the amount of sunlight striking the roof using dot products to determine how much solar power could be generated by solar panels on the roof. [Grade: 10-12 | Topics: vectors; dot and cross product; normal vectors; unit conversions ] (PDF)

Problem 652: VAP- Telemetry Math
Students work with data rates for the spacecraft and determine how much data needs to be stored. [Grade: 6-8 | Topics: megabytes; rates in time] (PDF)

Problem 651: VAP- The RBSP Satellite: Working with Octagons
Students work with the area formula for squares, rectangles and triangles to find the surface area of an octagonal satellite. [Grade: 9-12 | Topics: areas of simple figures; algebraic manipulation] (PDF)

Problem 650: VAP- Working with Areas of Rectangles and Circles
Students use the formulas for simple rectangle and circle areas to determine the areas of the holes in a satellite panel. [Grade: 3-5 | Topics: area of a rectangle; area of a circle] (PDF)

Problem 472: Investigating Juno's Elliptical Transfer Orbit
Students use the Standard Formula for an ellipse to study the elliptical orbit of the Juno spacecraft, and relate specific properties of the ellipse to features of the spacecrafts trajectory such as aphelion, perihelion, and ellipticity. [Grade: 9-12 | Topics: formula for an ellipse; semi-major and minor axis] (PDF)

Problem 471: Investigating the Launch of the Juno Spacecraft
Students use a series of images from a launch video to determine the scale of each image and determine the speed of the rocket as it leaves the gantry. [Grade: 6-8 | Topics: scale models; speed = distance/times] (PDF)

Problem 470: The Launch of the Juno Spacecraft - Ascent to orbit
Students use tabulated altitude and range data following the launch of the Juno mission, to determine the speed of the rocket as it travels to arth orbit. [Grade: 6-8 | Topics: scale models; speed = distance/time] (PDF)

Problem 469: Solar Energy and the Distance of Juno from the Sun
Students use the formula for an ellipse, along with the inverse-square law to create a mathemartical model that predicts the declining solar power produced by Junos solar panels as the spacecraft travels from Earth to Jupiter. [Grade: 9-12 | Topics: algebra; trigonometry; distance formula] (PDF)

Problem 457: The Interplanetary Voyage of MSL
Students use the properties of ellipses to determine the formula for the Hohmann Transfer Orbit taking the Mars Science Laboratory to Mars in 2012 [Grade: 10-11 | Topics: time=distance/speed; scale models; metric math; properties of ellipses] (PDF)

Problem 456: The Launch of the Mars Science Laboratory (MSL) in 2011
Students use a sequence of launch images to determine the Atlas V's launch speed and acceleration. By determining the scale of each image, they estimate average speeds during the first 4 seconds after lift-off. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 455: The Night Launch of STEREO in 2006
An example of old news seen in a different way! Students use a spectacular time-lapse photo of the launch of the STEREO mission obtained by photographer Dominic Agostini in 2006 to study parabolic curves. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math; equation of a parabola; curve fitting] (PDF)

Problem 419: The Space Shuttle: Fly me to the moon? Students discuss the popular misconception that the Space Shuttle can travel to the moon by examining the required orbit speed change and the capacity of the Shuttle engines to provide the necessary speed changes. [Grade: 6-8 | Topics: amount = rate x time ] (PDF)

Problem 324: Deep Impact Comet Flyby The Deep Impact spacecraft flew by the Comet Tempel-1 in 2005. Students determine the form of a function that predicts the changing apparent size of the comet as viewed from the spacecraft along its trajectory. [Grade: 9-12 | Topics: Algebra, geometry, differential calculus]

Problem 316: Counting Craters on the Hubble Space Telescope Students count craters on a piece of the Wide Field Planetary Camera recovered from the Hubble Space Telescope in 2009. They determine the cratering rate and use this to predict how many impacts the solar panels on the International Space Station experiences each day. [Grade: 6-9 | Topics: Counting; Area; density]

Problem 282: Exploring the Ares 1-X Launch: The Hard Climb to Orbit Students learn about the energy required to send a payload into orbit by studying the Ares 1-X rocket launch. [Grade: 8-10 | Topics: Algebra II]

Problem 281: Exploring the Ares 1-X Launch: Energy Changes Students learn about kinetic and potential energy while studying the Ares 1-X rocket launch. [Grade: 8-10 | Topics: Algebra II]

Problem 280: Exploring the Ares 1-X Launch: Parametrics Students learn about parametric equations to determine the path of the Ares 1-X rocket. [Grade: 8-10 | Topics: Algebra II; Parametric Equations]

Problem 279: Exploring the Ares 1-X Launch: Downrange Distance Students learn about the path of the Ares 1-X test launch and calculate its downrange landing distance in the Atlantic Ocean. [Grade: 8-10 | Topics: Algebra; Significant Figures; Metric to English Conversion]

Problem 276: Solid Rocket Boosters and Thrust Students learn how solid rocket boosters work, and calculate the SRB Thrust Curve using a simple geometric model and 'counting squares'. [Grade: 8-10 | Topics: Geometry, Cylindrical volumes and surface areas, Graphing data]

Problem 266: The Ares-V Cargo Rocket Students work with the equations for thrust and fuel loss to determine the acceleration curve of the Ares-v during launch. [Grade: 11-12 | Topics: Algebra II, properties of functions, differential calculus, Excel Spreadsheet]

Problem 260: Some Famous Unit Conversion Errors Students examine three famous unit conversion errors that led to catastrophic failures and near-death experiences. [Grade: 6-8 | Topics: unit conversion, metric measure]

Problem 245: Solid Rocket Boosters Students learn how SRBs actually create thrust, and study the Ares-V booster to estimate its thrust. [Grade: 8-10 | Topics: volume, area, unit conversions]

Problem 243: ISS - Orbit Altitude Changes Students read an essay describing the increases and decreases in the International Space Station orbit, and calculate the final orbit altitude after all the changes are applied. [Grade: 8-10 | Topics: combining positive and negative mixed numbers; fractions]

Problem 238: Satellite Drag and the Hubble Space Telescope Satellite experience drag with the atmosphere, which eventually causes them to burn up in the atmosphere. Students study various forecasts of the althtiude of the Hubble Space Telescope to estimate its re-entry year. [Grade: 8-10 | Topics: interpreting graphical data; predicting trends]

Problem 210: The Mathematics of Ion Rocket Engines- Students learn about the basic physics of ion engines, calculating speeds. [Grade: 9-12| Topics: Scientific Notation; Algebra II; evaluating formulae.]

Problem 206: Can You Hear me now? - Students learn about how the transmission of data is affected by how far away a satellite is, for a variety of spacecraft in the solar system [Grade: 6-8| Topics: multiplication; division; decimal numbers.]

Problem 203: Light Travel Times- Students determine the time it takes light to reach various objects in space. [Grade: 6-8| Topics: Scientific Notation; Multiplication; time = distance/speed.]

Problem 202: The Dawn Mission - Ion Rockets and Spiral Orbits- Students determine the shape of the trajectory taken by a spacecraft using a constant-thrust ion motor using differential and integral calculus for arc lengths. [Grade: 9-12| Topics: Calculus - Arc lengths.]

Problem 201: Fly Me To the Moon!- Students learn some basic principles and terminology about how spacecraft change their orbits to get to the moon. [Grade: 6-8| Topics: speed = distance/time; Pythagorean Theorem]

Problem 185: The International Space Station: Follow that graph!- Students use a plot of the orbit altitude of the ISS to pblackict its re-entry year after the peak of the next solar activity cycle. [Grade: 6-8| Topics: extrapolating a simple graph; estimation; forecasting]

Problem 181: Extracting Oxygen from Moon Rocks- Students use a chemical equation to estimate how much oxygen can be liberated from a sample of lunar soil. [Grade: 9-11| Topics: ratios; scientific notation; unit conversions]

Problem 179: Is There a Lunar Meteorite Impact Hazard? - Students work with areas, probability and impact rates to estimate whether lunar colonists are in danger of meteorite hazards. [Grade: 5-7| Topics: Area; unit conversions; rates]

Problem 166: The Dollars and Cents of Research - Students work with dollar amounts, hourly salary rates, percentages to explore various models of the cost of scientific research as seen by the individual scientist. [Grade: 4-6 | Topics: percentages, decimal math, simple rates (e.g dollars/hour)]

Problem 157: Space Shuttle Launch Trajectory - I - Students use the parametric equation for the altitude and range for an actual Shuttle launch to determine the speed and acceleration of the Shuttle during launch and orbit insertionh [Grade: 11-12 | Topics: Algebra; Calculus; Parametric Equations; Differentiation

Problem 133 How Big is It? - The Mars Rover. Students work with an image taken by the Mars Orbiter satellite of the Spirit landing site. They determine the image scale, and calculate the sizes of various surface features from the image. [Grade: 4 - 7 | Topics:image scaling; multiply, divide, work with millimeter ruler]

Problem 121 Ice on Mercury? Since the 1990's, radio astronomers have mapped Mercury. An outstanding curiosity is that in the polar regions, some craters appear to have 'anomalous reflectivity' in the shadowed areas of these craters. One interpretation is that this is caused by sub-surface ice. The MESSENGER spacecraft hopes to explore this issue in the next few years. In this activity, students will measure the surface areas of these potential ice deposits an calculate the volume of water that they imply. [Grade: 8-10 | Topics:Area of a circle; volume, density, unit conversion]

Problem 113 NASA Juggles Four Satellites at Once! Students will learn about NASA's Magnetospheric Multi-Scale (MMS) satellite mission, and how it will use four satellites flying in formation to investigate the mysterious process called Magnetic Reconnection that causes changes in Earth's magnetic field. These changes lead to the production of the Northern and Southern Lights and other phenomena. From the volume formula for a tetrahedron, they will calculate the volume of several satellite configurations and estimate the magnetic energy and travel times for the particles being studied by MMS. [Grade: 8-10 | Topics: Formulas with two variables; scientific notation]

Problem 108 A Problem in Satellite Synchrony The THEMIS program uses five satellites in five different orbits to study Earth's magnetic field and its changes during a storm. This problem asks students to use the periods of the five satellites to figure out when all 5 satellites will be lined-up as seen from Earth. They will do this by finding the Greatest Common Multiple of the five orbit periods, first for the case of 2 or 3 satellites, which can be easily diagrammed with concentric circles, then the case for all five satellites together. [Grade: 5-9 | Topics:multiplication; Greatest Common Multiple]

Problem 96 Hinode Satellite Power - Students will study the design of the Hinode solar satellite and calculate how much power it can generate from its solar panels. [Grade level: 6-8 | Topics:area of rectangle,area of cylinder, unit conversion]

Problem 95 A Study on Astronaut Radiation Dosages in SPace - Students will examine a graph of the astronaut radiation dosages for Space Shuttle flights, and estimate the total dosages for astronauts working on the International Space Station. [Grade level: 9-11 | Topics:Graph analysis, interpolation, unit conversion]

Problem 93 An Introduction to Radiation Shielding - Students calculate how much shielding a new satellite needs to replace the ISO research satellite. Students use a graph of the wall thickness versus dosage, and determine how thick the walls of a hollow cubical satellite have to be to blackuce the radiation exposure of its electronics. Students calculate the mass of the satellite and the cost savings by using different shielding. [Grade level: 9-11 | Topics: Algebra; Volume of a hollow cube; unit conversion]

Problem 89 Atmospheric Shielding from Radiation- III - This is Part III of a 3-part problem on atmospheric shielding. Students use exponential functions to model the density of a planetary atmosphere, then evaluate a definite integral to calculate the total radiation shielding in the zenith (straight overhead) direction for Earth and Mars. [Grade level: 11-12 | Topics: Evaluating an integral, working with exponential functions]

Problem 88 Atmospheric Shielding from Radiation- II - This is the second of a three-part problem dealing with atmospheric shielding. Students use the formula they derived in Part I, to calculate the radiation dosage for radiation arriving from straight overhead, and from the horizon. Students also calculate the 'zenith' shielding from the surface of Mars. [Grade level: 9-11 | Topics: Algebra I; evaluating a function for specific values]

Problem 87 Atmospheric Shielding from Radiation- I - This is the first part of a three-part problem series that has students calculate how much radiation shielding Earth's atmosphere provides. In this problem, students have to use the relevant geometry in the diagram to determine the algebraic formula for the path length through the atmosphere from a given location and altitude above Earth's surface. [Grade level: 9-11 | Topics: Algebra II, trigonometry]

Problem 83 Luner Meteorite Impact Risks - In 2006, scientists identified 12 flashes of light on the moon that were probably meteorite impacts. They estimated that these meteorites were probably about the size of a grapefruit. How long would lunar colonists have to wait before seeing such a flash within their horizon? Students will use an area and probability calculation to discover the average waiting time. [Grade level: 8-10 | Topics: arithmetic; unit conversions; surface area of a sphere) ]

Problem 80 Data Corruption by High Energy Particles - Students will see how solar flares can corrupt satellite data, and create a timeline for a spectacular episode of data loss recorded by the SOHO satellite using images obtained by the satellite. Students will also calculate the speed of the event as particles are ejected from the sun and streak towards earth. [Grade level: 6-8 | Topics: Time and speed calculations; interpreting scientific data ]

Problem 79 Correcting Bad Data Using Partity Bits - Students will see how computer data is protected from damage by radiation 'glitches' using a simple error-detection method involving the parity bit. They will reconstruct an uncorrupted sequence of data by checking the '8th bit' to see if the transmitted data word has been corrupted. By comparing copies of the data sent at different times, they will reconstruct the uncorrupted data. [Grade level: 4-6 | Topics: addition, subtraction, comparing the numbers 1 and 0 ]

Problem 76 Radon Gas in the Basement - This problem introduces students to a common radiation problem in our homes. From a map of the United States provided by the US EPA, students convert radon gas risks into annual dosages. [Grade level: 6-8 | Topics: Unit conversion, arithmetic operations]

Problem 36 The Space Station Orbit Decay and Space Weather Students will learn about the continued decay of the orbit of the International Space Station by studying a graph of the Station's altitude versus time. They will calculate the orbit decay rates, and investigate why this might be happening. [Grade: 5 - 8 | Topics: Interpreting graphical data; decimal math]

Problem 8 Making a Model Planet Students use the formula for a sphere, and the concept of density, to make a mathematical model of a planet based on its mass, radius and the density of several possible materials (ice, silicate rock, iron, basalt). [Grade: 7 - 9 | Topics: Volume of sphere; mass = density x volume; decimal math; scientific notation]

## Rocketry, Launches, Launch Vehicles

Problem 618: Voyager 1 Begins its Interstellar Journey
Students calculate speeds, light travel times and density for the Voyager 1 encounter with the interstellar medium beyond the orbit of Pluto. [Grade: 6-8 | Topics: speed=distance/time; time = distance/speed; number = density x volume] (PDF)

Problem 617: The Basic Mathematics of Rocket Engines - I
Students learn about and calculate specific impulse and thrust. [Grade: 9-12 | Topics: proportions; unit conversions; rates of change] (PDF)

Problem 616: Adding a Level Gauge to a Conical Tank
Students work with the formula for the volume of a cone [Grade: 6-8 | Topics:geometry; volume of a cone ] (PDF)

Problem 614: A Simple Gauge in a Tank - II
Students work with the formula for the volume of a conical solid to design a gas tank gauge. [Grade: 6-8 | Topics: geometry; volume of a cone; tangents ] (PDF)

Problem 610: Calculating the Volume of the J-2x Rocket Engine Bell
Students explore conical volumes by examining the diemnsions of a large rocket engine. [Grade: 6-8 | Topics:volume of a conical solid; solving a quadratic equation for a selected value of X] (PDF)

Problem 609: Volumes of Solids - Packing for a trip to the Moon
Students calculate volumes of rectangular solids and pack a volume-limited travel kit used by astronauts. [Grade: 3-5 | Topics: volume of rectangular solids; integers] (PDF)

Problem 607: The Launch of LADEE to the Moon
Students plot the altitude, range and speed of the LADEE rocket launch and investigate rates of change including acceleration by graphing the tabular data and determining the slope of the graph using the definition of the slope of a line between two points. [Grade: 6-8 | Topics: Graphing tabular data; determining the slope of a line; rates of change] (PDF)

Problem 601: Gravity and Escape Speed
Students calculate the escape speed for various planets using a simple 'square root' equation. [Grade: 6-8 | Topics: evaluating functions; square-roots ] (PDF)

Problem 600: Reading a Speed vs Time Graph - acceleration
Students read a graph to determine how speed is related to acceleration as the area under a curve. [Grade: 9-12 | Topics:areas of rectangles and triangles; interpreting graphical data ] (PDF)

Problem 599: Exploring Artificial Gravity
Students work with centrifugal forces to calculate the acceleration of County Fair rides; rotating spacecraft and the acceleration of rockets to see if artificial gravity can be created. [Grade: 9-12 | Topics: evaluating functions] (PDF)

Problem 598: Gravity and Energy
Students the energy equation E = mgh to expore the energy of falling water near Earths surface to calculate the energy of a water fall. [Grade: 6-8 | Topics: evaluating functions; unit conversions] (PDF)

Problem 597: The Physics of Rock Throwing
Students study the motion of a thrown rock to explore the parabolic shape of the rocks motion. [Grade: 9-12 | Topics: quadratic equations; parametric equations; parabolas ] (PDF)

Problem 596: Distance Traveled Under Free Fall by Gravity
Students explore accelerated motion and distance traveled using an equation that related distance to time-squared, and solve the equation under various conditions. [Grade: 6-8 | Topics: solving for X; quadratic monomials; square roots] (PDF)

Problem 595: Gravity and Falling Bodies
Students work with simple linear equations to study the speed of falling bodies under gravity. [Grade: 6-8 | Topics:linear equations; solving for x ] (PDF)

Problem 507: Exploring the Launch of the Falcon 9
Students use data from the launch of the Falcon 9 booster to determine its speed and acceleration. [Grade: 6-8 | Topics: speed=distance/time; Time calculations] (PDF)

Problem 471: Investigating the Launch of the Juno Spacecraft
Students use a series of images from a launch video to determine the scale of each image and determine the speed of the rocket as it leaves the gantry. [Grade: 6-8 | Topics: scale models; speed = distance/times] (PDF)

Problem 470: The Launch of the Juno Spacecraft - Ascent to orbit
Students use tabulated altitude and range data following the launch of the Juno mission, to determine the speed of the rocket as it travels to arth orbit. [Grade: 6-8 | Topics: scale models; speed = distance/time] (PDF)

Problem 456: The Launch of the Mars Science Laboratory (MSL) in 2011
Students use a sequence of launch images to determine the Atlas V's launch speed and acceleration. By determining the scale of each image, they estimate average speeds during the first 4 seconds after lift-off. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math] (PDF)

Problem 455: The Night Launch of STEREO in 2006
An example of old news seen in a different way! Students use a spectacular time-lapse photo of the launch of the STEREO mission obtained by photographer Dominic Agostini in 2006 to study parabolic curves. [Grade: 8-10 | Topics: time=distance/speed; scale models; metric math; equation of a parabola; curve fitting] (PDF)

Problem 442: Modeling the Atmospheric Re-entry of UARS
Students graph the altitude of the UARS satellite in the weeks before re-enrty to explore the accelerating effects of atmospheric drag. They create a mathematical model that fits the data, and use this to make their own prediction of the re-entry date. [Grade: 8 - 11 | Topics: graphing data; linear equations; exponential and power functions] (PDF)

Problem 438: The Last Flight of the Space Shuttle Endeavor
Students use tabular data and graphing to determine the launch speed and acceleration of the Space Shuttle from the launch pad. [Grade: 6-8 | Topics: tabular data, graphing, metric measurement, speed=distance/time] (Questions Only)

Problem 437: Saturn V Rocket Launch Speed and Height
Students tabular data to determine the launch speed of the Saturn V rocket from the launch pad. [Grade: 6-8 | Topics: tabular data, graphing, metric measurement, speed=distance/time] (PDF)

Problem 436: Space Shuttle Challenger Deploys the INSAT-1B Satellite
Students use a sequence of images to determine the launch speed of the satellite from the Space Shuttle cargo bay. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 435: Apollo-17 Lanuch from Lunar Surface
Students use a sequence of images to determine the speed of ascent of the Apollo-17 capsule from the lunar surface. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 433: Space Shuttle Atlantis - Plume Speed
Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 432: Space Shuttle Atlantis - Exhaust Speed
Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 431: Space Shuttle Atlantis - Launch Speed
Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 430: Space Shuttle Atlantis - Ascent to Orbit
Students use a sequence of images from a video of the launch to determine speed from the time interval between the images, and the scale of each image. [Grade: 6-8 | Topics: scale, metric measurement, speed=distance/time] (PDF)

Problem 419: The Space Shuttle: Fly me to the moon? Students discuss the popular misconception that the Space Shuttle can travel to the moon by examining the required orbit speed change and the capacity of the Shuttle engines to provide the necessary speed changes. [Grade: 6-8 | Topics: amount = rate x time ] (PDF)

Problem 282: Exploring the Ares 1-X Launch: The Hard Climb to Orbit Students learn about the energy required to send a payload into orbit by studying the Ares 1-X rocket launch. [Grade: 8-10 | Topics: Algebra II]

Problem 281: Exploring the Ares 1-X Launch: Energy Changes Students learn about kinetic and potential energy while studying the Ares 1-X rocket launch. [Grade: 8-10 | Topics: Algebra II]

Problem 280: Exploring the Ares 1-X Launch: Parametrics Students learn about parametric equations to determine the path of the Ares 1-X rocket. [Grade: 8-10 | Topics: Algebra II; Parametric Equations]

Problem 279: Exploring the Ares 1-X Launch: Downrange Distance Students learn about the path of the Ares 1-X test launch and calculate its downrange landing distance in the Atlantic Ocean. [Grade: 8-10 | Topics: Algebra; Significant Figures; Metric to English Conversion]

Problem 277: Deep Impact Comet Encounter Students learn about the Deep Impact experiment involving Comet Tempel-1, and how the path of an asteroid can be changed by using the Law of Conservation of Momentum. [Grade: 10-12 | Topics: Algebra; Scientific Notation; distance = speedxtime]

Problem 276: Solid Rocket Boosters and Thrust Students learn how solid rocket boosters work, and calculate the SRB Thrust Curve using a simple geometric model and 'counting squares'.. [Grade: 8-10 | Topics: Geometry, Cylindrical volumes and surface areas, Graphing data]

Problem 266: The Ares-V Cargo Rocket Students work with the equations for thrust and fuel loss to determine the acceleration curve of the Ares-v during launch. [Grade: 11-12 | Topics: Algebra II, properties of functions, differential calculus, Excel Spreadsheet]

Problem 245: Solid Rocket Boosters Students learn how SRBs actually create thrust, and study the Ares-V booster to estimate its thrust. [Grade: 8-10 | Topics: volume, area, unit conversions]

Problem 243: ISS - Orbit Altitude Changes Students read an essay describing the increases and decreases in the International Space Station orbit, and calculate the final orbit altitude after all the changes are applied. [Grade: 8-10 | Topics: combining positive and negative mixed numbers; fractions]

Problem 238: Satellite Drag and the Hubble Space Telescope Satellite experience drag with the atmosphere, which eventually causes them to burn up in the atmosphere. Students study various forecasts of the althtiude of the Hubble Space Telescope to estimate its re-entry year. [Grade: 8-10 | Topics: interpreting graphical data; predicting trends]

Problem 210: The Mathematics of Ion Rocket Engines- Students learn about the basic physics of ion engines, calculating speeds. [Grade: 9-12| Topics: Scientific Notation; Algebra II; evaluating formulae ]

Problem 202: The Dawn Mission - Ion Rockets and Spiral Orbits- Students determine the shape of the trajectory taken by a spacecraft using a constant-thrust ion motor using differential and integral calculus for arc lengths. [Grade: 9-12| Topics: Calculus - Arc lengths ]

Problem 201: Fly Me To the Moon!- Students learn some basic principles and terminology about how spacecraft change their orbits to get to the moon. [Grade: 6-8| Topics: speed = distance/time; Pythagorean Theorem]

Problem 185: The International Space Station: Follow that graph!- Students use a plot of the orbit altitude of the ISS to pblackict its re-entry year after the peak of the next solar activity cycle. [Grade: 6-8| Topics: extrapolating a simple graph; estimation; forecasting]

Problem 157: Space Shuttle Launch Trajectory - I - Students use the parametric equation for the altitude and range for an actual Shuttle launch to determine the speed and acceleration of the Shuttle during launch and orbit insertion [Grade: 11-12 | Topics: Algebra; Calculus; Parametric Equations; Differentiation

## Telescopes, Design, Remote Sensing

Students compare several telescopes and select the one with the best performance and lowest cost. [Grade: 6-8 | Topics: simple ratio formula; decimal math] (PDF)

Problem 582: How do Telescopes Magnify?
Students use a simple ratio formula to calculatethe magnification of a telescope. [Grade: 3-5 | Topics: division of two decimal numbers; evaluating simple ratios. ] (PDF)

Problem 581: How Telescopes Work
Students compare hwo much light a telescope can gather compared to the human eye. [Grade: 3-5 | Topics: Area of a circle ] (PDF)

Problem 579: The Scale of an Image with a Telescope
Students desigh digital cameras for telescopes given information about the image scale of the telescope and the pixel dimensions. [Grade: 6-8 | Topics: area; evaluating simple equations; unit conversions] (PDF)

Problem 578: Digital Camera Math
Students learn about digital cameras and how to interpret formats, megapixels and angular resolution. [Grade: 6-8 | Topics: integer math; area of a square] (PDF)

Problem 577: Designing a Telescope System
Students design two telescopes given information about the desired properties for conducting research. [Grade: 6-8 | Topics: graphing inequalities; evaluating simple equations ] (PDF)

Problem 576: Telescope Resolution - How much detail can you see?
Students determine the resolving power of a telescope and the limit to the finest details that can be see for a telescope of a specific diameter. [Grade: 6-8 | Topics: Angular measure; arcseconds; simple equations ] (PDF)

Problem 575: Telescope Field of View - How much can you see?
Students calculate the angular field of view for various telescopes using a simple formula of the form: F = A/B [Grade: 6-8 | Topics: Angular measure; degrees; ] (PDF)

Problem 574: Telescope Light Gathering Ability - Seeing Faint Stars
Students calculate the light gathering ability of various telescopes compared to the human eye. [Grade: 6-8 | Topics: Area of a circle ] (PDF)

Problem 573: Calculating the Magnification of a Telescope
Students fill in missing numbers in a table using proportions and evaluating a simple equation for magnification. [Grade: 6-8 | Topics: proportions] (PDF)

Problem 571: Focal Lengths, Apertures and F/numbers
Students learn about the basic terms that define the performance of a digital camera or a telescope. [Grade: 6-8 | Topics: fractions; integer division; evaluating simple equations ] (PDF)

Problem 567:Exploring Parabolas - The shape of a satellite dish
Students use the equation for a parabola to determine the focus location for a solar cooker and a sound amplifier dish given their diameters and depths. [Grade: 9-12 | Topics: geometry; properties of parabolas] (PDF)

Problem 566:Exploring Light Brightness and the Inverse Square Law
Students collect data and explore the inverse square law using a light meter. They deduce the formula for the brightness of a lamp given its distance and wattage. [Grade: 6-8 | Topics: graphing tabular data; surface area of a sphere; ] (PDF)

Problem 565:Mapping Earth from Space - Swaths and Coverage
Students explore how satellite observing swaths add up to give full coverage of earths surface. [Grade: 6-8 | Topics: geometry;scale model; working with square roots ] (PDF)

Problem 524:Seeing the Martian Surface with IDC
Students learn about the IDC camera and calculate resolution and how many images are needed to map the InSight landing area. [Grade: 6-8 | Topics: ANgular measurfe, degrees and seconds; image scal; tiling an area with overlap. ] (PDF)

Problem 256: A High-resolution Satellite Photo Students examine a satelite photo of the Tennessee Court House from the GEO-1 satellite and determine the sizes of familiar features in the image. [Grade: 6-8 | Topics: scale, ratios, proportions' angle measure, triangle geometry]

Problem 250: The Most Important Equation in Astronomy Students learn about how an instrument's ability to see details depends on its size and its operating wavelength - the key to designing any telescope or camera. [Grade: 8-10 | Topics: geometry, angle measure, scientific notation]

## Orbits

Problem 678:VAB-Navigating a Magnetic Field with Vector Dot Products! Students work with vectors to determine a spacecrafts orientation relative to Earths magnetic field. They compute the expected strength of the magnetic field parallel and perpendicular to the spacecraft motion vector. [Grade: 10-12 | Topics: vectors; dot products; vector projections ] (PDF)

Problem 677:VAB-Navigating in a Magnetic Field Using Linear Equations Students model spacecraft motion and the local magnetic field direction using two linear equations, then determine the line perpendicular to the spacecraft motion and the angle of motion relative to the magnetic field. [Grade: 8-10 | Topics: graphing linear equations; equation of line perpendicular to another line; geometry ] (PDF)

Problem 676:VAB-Navigating in a Magnetic World! Students explore how some satellites navigate in space using Earth's magnetic field and its orientation to the spacecraft. [Grade: 7-10 | Topics: coordinate plotting; geometry; graphing data ] (PDF)

Problem 675:VAB-Earth�s Magnetic Field and the Van Allen Probes Students explore Earth's magnetic field as seen by spacecraft in their orbit to determine the local 'compass direction' of the magnetic field. [Grade: 7-10 | Topics: coordinate plotting; properties of right triangles; Pythagorean theorem] (PDF)

Problem 674:VAB - Exploring the Orbit of the Van Allen Probes Students explore the speed of the spacecraft in their orbit around Earth using coordinate graphing, time differences, pythagorean theorem and unit conversions. [Grade: 7-10 | Topics: coordinate plotting; unit conversion; rates ] (PDF)

Problem 472: Investigating Juno's Elliptical Transfer Orbit
Students use the Standard Formula for an ellipse to study the elliptical orbit of the Juno spacecraft, and relate specific properties of the ellipse to features of the spacecrafts trajectory such as aphelion, perihelion, and ellipticity. [Grade: 9-12 | Topics: formula for an ellipse; semi-major and minor axis] (PDF)

Problem 457: The Interplanetary Voyage of MSL
Students use the properties of ellipses to determine the formula for the Hohmann Transfer Orbit taking the Mars Science Laboratory to Mars in 2012 [Grade: 10-11 | Topics: time=distance/speed; scale models; metric math; properties of ellipses] (PDF)

## Data, Telemetry, Transmission

Problem 567:Exploring Parabolas - The shape of a satellite dish
Students use the equation for a parabola to determine the focus location for a solar cooker and a sound amplifier dish given their diameters and depths. [Grade: 9-12 | Topics: geometry; properties of parabolas] (PDF)

Problem 525:Exploring the InSight Lander Telemetry Data Flow
Students explore how long it takes to transmit digital data using examples from downloading songs from their computer to their ipod. [Grade: 6-8 | Topics: working with kilo, mega and rates of data transfer in bytes/sec. ] (PDF)

Problem 235: Scientific Data: The gift that keeps on giving! Students learn about gigabytes and terrabytes of data and the rates of data generation by NASA missions and how to store it. [Grade: 6-8 | Topics: metric units; rates; money]

Problem 207: The STEREO Mission: getting the message across- Students learn about how the transmission of data is affected by how far away a satellite is for the two satellites in the STEREO constellation. [Grade: 6-8| Topics: multiplication; division; decimal numbers ]

Problem 206: Can You Hear me now? - Students learn about how the transmission of data is affected by how far away a satellite is, for a variety of spacecraft in the solar system [Grade: 6-8| Topics: multiplication; division; decimal numbers ]

Problem 79 Correcting Bad Data Using Partity Bits - Students will see how computer data is protected from damage by radiation 'glitches' using a simple error-detection method involving the parity bit. They will reconstruct an uncorrupted sequence of data by checking the '8th bit' to see if the transmitted data word has been corrupted. By comparing copies of the data sent at different times, they will reconstruct the uncorrupted data. [Grade level: 4-6 | Topics: addition, subtraction, comparing the numbers 1 and 0 ]

Problem 69 Single Event Upsets in Aircraft Avionics - Radiation is problem for high-altitude commercial and research aircraft. Showers of high-energy neutrons cause glitches in computer electronics and other aircraft systems. This problem investigates the neutron background radiation at 30,000 to 100,000 feet based on actual flight data, and has students calculate how many computer memory glitches will happen over a set amount of flight time. [Grade level: 8-10 | Topics: decimals, unit conversions, graph analysis]

Problem 80 Data Corruption by High Energy Particles - Students will see how solar flares can corrupt satellite data, and create a timeline for a spectacular episode of data loss recorded by the SOHO satellite using images obtained by the satellite. Students will also calculate the speed of the event as particles are ejected from the sun and streak towards earth. [Grade level: 6-8 | Topics: Time and speed calculations; interpreting scientific data ]

## Mission Planning

Problem 679:Shipping Cargo to the International Space Station Students work with kilograms and watts to determine the cargo capacity for re-supply missions to the International Space Station. [Grade: 4-6 | Topics: addition, division, multiplication of 3 and 4-digit numbers. ] (PDF)

Problem 536:Exploring a Possible InSight Landing Area on Mars
Students work with latitude and longitude and scaled images of mars to locate the InSight proposed landing area, and describe the terrain of the landing area. [Grade: 6-8 | Topics: degree measure; latitude and longitude; working with scaled images; metric measure] (PDF)

Problem 535:Comparing the InSight Landing Area to a City Block!
Students use scaled images of a proposed InSIght landing area and a scaled image of an urban neighborhood on Earth to compare the sizes of familiar things with the unfamiliar martian landscape. [Grade: 6-8 | Topics: scale; proportion; metric measurement] (PDF)

Problem 532:The Distance to the Martian Horizon
Students devive a basic equation for the distance to the horizon on a spherical body using the Pythagorean Theorem and a bit of algebra. The estimate the number of cell towers needed to cover Mars. [Grade: 8-10 | Topics: Pythagorean Theorem, Algebra; scientific notation; areas of spheres and circles ] (PDF)

Problem 527:Exploring Heat Flow and Insulation
Students explore how insulation works to reduce heat flow. They convert a verbal description of a formula expressed in proportions, and use it to calculate why aluminum pots heat faster than steel pots, and how we can determine the properties of martian sooil from heat flow and temperature changes. [Grade: 8-10 | Topics: algebra; rates of change ] (PDF)

Problem 523:Telling Time on Mars - Earth Days and Mars Sols
Students work with two clocks on Earth and Mars and learn about earth and mars time given that a day on Mars is 40 minutes longer than an Earth day. [Grade: 6-8 | Topics: time calculations, hours, minutes, seconds; length of day ] (PDF)

Problem 522:Radio Communications with Earth ? The Earth-Sun Angle
The earth-sun angle is given in tabular form in degrees. Students graph the data and find the dates when transmissions to Earth cannot occur. [Grade: 8-10 | Topics: Interpreting tabular data; rates and slopes ] (PDF)

Problem 520:The Work Area In Front of the Lander
Students estimate the area in front of the InSight lander where experiments will be conducted and instruments moved with a single robotic arm. [Grade: 6-8 | Topics: Area of a circle segment; Area common to two intersecting circles] (PDF)

Problem 519:Scheduling Events in Time for Launch
Students learn about scheduling many events along a timeline (breakfast, packing, driving, etc ) by planning a family trip where the family members have to arrive at the airport for a flight that leaves at a specific date and time. [Grade: 5-7 | Topics: working with time units; creating a timeline] (PDF)

Problem 373: Solar Probe Plus - Having a hot time near the sun!
Students use a simple equation to estimate the temperature reached by the Solar Probe Plus spacecraft as it gets close to the sun. [Grade: 8-10 | Topics: Evaluating a function; square roots and forth roots] (PDF)

Problem 208: Optimization- Students determine the optimal dimensions of an hexagonal satellite to maximize its surface area given its desiblack volume. [Grade: 9-12| Topics: Calculus; differentiation ]

Problem 108 THEMIS - A Problem in Satellite Synchrony - The THEMIS program uses five satellites in five different orbits to study Earth's magnetic field and its changes during a storm. This problem asks students to use the periods of the five satellites to figure out when all 5 satellites will be lined-up as seen from Earth. They will do this by finding the Greatest Common Multiple of the five orbit periods, first for the case of 2 or 3 satellites, which can be easily diagrammed with concentric circles, then the case for all five satellites together. [Grade level: 5-9 | Topics:multiplication; Greatest Common Multiple]

Problem 84 How Far is the Horizon? - Students use geometry, and the Pythagorean Theorem, to determine the formula for the distance to the horizon on any planet with a radius, R, from a height, h, above its surface. Additional problems added that involve calculus to determine the rate-of-change of the horizon distance as you change your height. [Grade level: 9-11 | Topics: Algebra, Pythagorean Theorem; Experts: DIfferential calculus) ]

## Spacecraft Design, Radiation and Power Generation

Problem 673:VAB - An Improved Model for Van Allen Belt Radiation Dose Students use a detailed model of the path of a satellite and the radiation dose rate along the path to calculate the total radiation dose to the spacecraft. [Grade: 9-11 | Topics: Polynomial equations; trigonometric equations; composite finctions f(f(x)); estimating areas under curves] (PDF)

Problem 672:VAB - Modeling the Radiation Dose of the Van Allen Probes Students create a simple mathematical model of the radiation exposure to the VABP satellites as they travel through the Van Allen belts. [Grade: 11-12 | Topics: Parametric equations;composite functions f(g(x)); integral calculus ] (PDF)

Problem 671:VAB - The Van Allen Probes and Radiation Dose Students study radiation dose units and estimate the exposures for a human living on the gropund; an astronaut in the ISS, and the Van Allen belt environment. [Grade: 8-10 | Topics: Unit conversion; rates] (PDF)

Problem 518:The InSight Seismographic Station Solar Power System
Students explore the properties of decagons to determine the area of the solar panels used on the InSight lander. [Grade: 7-9 | Topics: area of regular polygons; estimating areas of non-square shapes] (PDF)

Problem 469: Solar Energy and the Distance of Juno from the Sun
Students use the formula for an ellipse, along with the inverse-square law to create a mathemartical model that predicts the declining solar power produced by Junos solar panels as the spacecraft travels from Earth to Jupiter. [Grade: 9-12 | Topics: algebra; trigonometry; distance formula] (PDF)

Problem 96 Hinode Satellite Power - Students will study the design of the Hinode solar satellite and calculate how much power it can generate from its solar panels. [Grade level: 6-8 | Topics:area of rectangle,area of cylinder, unit conversion]

Problem 41 Solar Energy in Space Students will calculate the area of a satellite's surface being used for solar cells from an actual photo of the IMAGE satellite. They will calculate the electrical power provided by this one panel. Students will have to calculate the area of an irregular region using nested rectangles. [Grade level: 6 - 8 | Topics: area of irregular polygon; decimal math]

Problem 38 Solar Eclipses and Satellite Power From the ground we see total solar eclipses where the New Moon passes directly between Earth and Sun. Satellites use solar cells to generate electricity, but this is only possible when the Earth is not 'eclipsing' the sun. Students will create a scaled drawing of the orbits of three satellites around Earth, and calculate how long each satellite will be in the shadow of Earth. They will be asked to figure out how to keep the satellites operating even without sunlight to power their solar panels. [Grade level: 7-9 | Topics: Geometry; time = distance/speed]

Problem 28 Satellite Power and Cosmic Rays Most satellites operate by using solar cells to generate electricity. But after years in orbit, these solar cells produce less electricity because of the steady impact of cosmic rays. In this activity, students read a graph that shows the electricity produced by a satellite's solar panels, and learn a valuable lesson about how to design satellites for long-term operation in space. [Grade level: 7-9 | Topics: Graph analysis; Area calculation, unit conversions, extrapolation]

Problem 16 Solar Power and Satellite Design Students perform simple surface area calculations to determine how much solar power a satellite can generate, compared to the satellite's needs. [Grade level: 7-9 | Topics: Area of irregular polygon; decimal math; unit conversions]

Problem 2 Satellite Surface Area Students calculate the surface area of an octagonal cylinder and calculate the power it would yield from solar cells covering its surface. [Grade level: 7-10 | Topics: Area of hexagonal cylinder; decimal math; unit conversions]

Problem 493: Fun with Gears and Fractions
Students learn about how simple fractions are used to describe gears and gear trains that reduce or increase speed. [Grade: 4-7 | Topics: multiplying simple fractions] (PDF)

Problem 463: A Simple Fuel Gauge in a Cylindrical Tank
Rockets use fuel tanks that can be approximated as cylinders. In this simple geometric exercise, students work the formula for the volume of a cylinder to add a fuel gauge at the right level to indicate how much fuel remains. [Grade: 7-9 | Topics: VOlume of cylinder; proportions] (PDF)

Problem 193: Fluid Level in a Spherical Tank - Students explore the relationship between volume, and the height of fluid in a spherical tank as fluid is being drained at a constant rate. [Grade: 10-12| Topics: Algebra, differential calculus, related rates]

Problem 459: A piece of history - space shuttle thermal tiles
Students explore volume density and mass using the Space Shuttle thermal tiles. Get your own free tile from NASA too! [Grade: 6-8 | Topics: mass = density x volume; metric conversion] (PDF)

Problem 95 A Study on Astronaut Radiation Dosages in SPace - Students will examine a graph of the astronaut radiation dosages for Space Shuttle flights, and estimate the total dosages for astronauts working on the International Space Station. [Grade level: 9-11 | Topics:Graph analysis, interpolation, unit conversion]

Problem 93 An Introduction to Radiation Shielding - Students calculate how much shielding a new satellite needs to replace the ISO research satellite. Students use a graph of the wall thickness versus dosage, and determine how thick the walls of a hollow cubical satellite have to be to blackuce the radiation exposure of its electronics. Students calculate the mass of the satellite and the cost savings by using different shielding. [Grade level: 9-11 | Topics: Algebra; Volume of a hollow cube; unit conversion]

Problem 82 Basic Unit Conversions - Students will use a number of obscure English units measures to convert from metric to English units and back, and answer some unusual questions! [Grade level: 6-8 | Topics: arithmetic; unit conversions involving 1 to 5 steps) ]

Problem 171: More Unit Conversions- Students work with four unit conversion problems that are a bit tricky! [Grade: 6-8 | Topics: unit conversions]

Problem 195: Advanced Unit Conversions - Students work with more unit conversions and use them to solve a series of practical problems in science and solar energy. [Grade: 6-10| Topics: unit conversions.]

Problem 260: Some Famous Unit Conversion Errors Students examine three famous unit conversion errors that led to catastrophic failures and near-death experiences. [Grade: 6-8 | Topics: unit conversion, metric measure]

Problem 246: Evaluating Secondary Physical Constants Students evaluate complicated algebraic quantities that define important constants in physics with both integer and fractional exponents. [Grade: 10-12 | Topics: Algebra; significant figures, scientific notation]