## Math problems featuring comets, asteroids and other objects

## Comet ISON

**Problem 584: Comparing Comets Up Close with NASA Spacecraft **

Students compare five comets and determine size ranges and percentages.
[Grade: 3-5 | Topics: percentages; volume of a cube]
(PDF)

**Problem 563:Comet ISON and its Close Encounter with Mars**

Students use tabular data to determine the date and time of closest approach to Mars
[Grade: 3-5 | Topics: graphing tabular data]
(PDF)

**Problem 562:Exploring the Orbit of Comet ISON**

Students use tabulated data to estimate when this comet will make its closest approach to the sun in 2013.
[Grade: 6-8 | Topics: graphing tabular data; scale; measurement; distance between points]
(PDF)

**Problem 560:The Orbit of Comet ISON**

Students explore how close Comet ISON will get to Mercury, Venus, Earth and Mars during its 2013 passage.
[Grade: 3-5 | Topics: Interpreting tabular data; graphing ]
(PDF)

**Problem 559:Comet ISON Losing Mass as it Approaches the Sun.**

Students estimage how much mass the comet will loose at its present rate.
[Grade: 6-8 | Topics: volume of a sphere; rates; mass=density x volume]
(PDF)

## Other Comets and Minor Bodies

**Problem 587: Comet Encounters after Discovery **

Students examine how often newly discovered comets approach Earth and become a hazard, and how soon after discovery these close passes can occur.
[Grade: 3-5 | Topics: Averaging, percentages]
(PDF)

**Problem 586: Searching for Comets **

Students use tabular data on the detection of new comets since 1999 to explore detection rates over time.
[Grade: 3-5 | Topics: Percentages]
(PDF)

**Problem 585: Exploring Comet Orbits **

Students explore the elliptical orbit of Halleys Comet and determine its period and the speed of the comet.
[Grade: 6-8 | Topics: speed=distance/time]
(PDF)

**Problem 558:How Quickly are Near Earth Objects Being Discovered?**

Students work with data presented in bar graphs to estimate how many more hazardous Near Earth Objects (NEOS) remein to be found.
[Grade: 6-8 | Topics:bar graphs ]
(PDF)

**Problem 557:The 10000th Near Earth Asteroid 2013 MZ5**

Students graph tabulated data to determine when this asteroid is closest to Earth and its speed at that time.
[Grade: 6-8 | Topics: grapohing tabuklar data; solving a linear equation]
(PDF)

**Problem 481: Pluto's Fifth Moon **

Students explore Kepler's Third Law and estimate the orbit period of a hypothetical sixth moon using the distance:period law.
They also determine the mass of Pluto using the orbit data, including the recently discovered fifth moon (P5) of Pluto by the Hubble Space Telescope.
[Grade: 9-12 | Topics: Power functions; integer exponents; Scientific Notation; tabular data]
(PDF)

**Problem 454: The Closest Approach of Asteroid 2005YU55 - III **

Students work with the equation of a circle and line to find the orbit intersection points, midpoint, and closest distance to earth.
[Grade: 8-10 | Topics: time=distance/speed; scale models; metric math]
(PDF)

**Problem 453: The Closest Approach of Asteroid 2005YU55 - II **

Students work with the properties of circles and angular measure to see where the moon will be at the start of the asteroid encounter.
[Grade: 6-8 | Topics: angular measure; time=distance/speed; scale models; metric math]
(PDF)

**Problem 452: The Closest Approach of Asteroid 2005YU55 - I **

Students work with a scaled drawing of the orbit of the moon and the asteroid trajectory to predict where the asteroid will be relative to earth and the orbit of the moon.
[Grade: 6-8 | Topics: time=distance/speed; scale models; metric math]
(PDF)

**Problem 434: Dawn Spacecraft Sees Asteroid Vesta Up-Close! **

Students use an image of the asteroid to determine the diameters of craters
and mountains using a millimeter ruler and the scale of the image in meters per millimeter.
[Grade: 6-8 | Topics: scale, metric measurement]
(PDF)

**Problem 387: A Mathematical Model of Water Loss from Comet Tempel-1 **
Students use data from the Deep Impact spacecraft to create a simple empirical model for predicting the rate of water loss from a comet based on actual data.
[Grade: 8-10 | Topics: graphing; fitting a parabola to data; evaluating functions]
(PDF)

**Problem 383: Estimating the mass of Comet Hartley 2 using calculus.**

Students use a recent image of the nucleus of Comet Hartley 2 taken by the Deep Impact/EPOXI camera and a shape function described
by a fourth-order polynomial to calculate the volume of the comet's head using integral calculus.
to estimate the volume of the comets nucleus, and its total mass,
[Grade: 12 | Topics: Volume integral using disk method; scale model; scientific notation; unit conversion]
(PDF)

**Problem 382: Estimating the mass and volume of Comet Hartley 2.**

Students use a recent image of the nucleus of Comet Hartley 2 taken by the Deep Impact/EPOXI camera and a simple geometric 'dumbell'
model based on a cylinder and two spheres, to estimate the volume of the comets nucleus, and its total mass.
[Grade: 8-10 | Topics: volume of a sphere and cylinder; scale model; scientific notation; unit conversion]
(PDF)

**Problem 377: Deep Impact: Approaching Comet Hartley-2 **

Students use data for the brightness of Comet Hartley-2 measured by the Deep Impact spacecraft to create a linear equation for its approach distance,
and use the inverse-square law to estimate its brightness on October 13, 2010.
[Grade: 8-10 | Topics: linear modeling from data; inverse-square law]
(PDF)

**Problem 374: Deep Impact - Closing In on Comet 103P/Hartley 2**

Students use the Tangent formula to figure out the angular size of the comet at closest approach, and the scale of the HRI camera image.
[Grade: 8-10 | Topics: Scaled images; trigonometry; angle measure]
(PDF)

**Problem 371: Close Encounters of the Asteroid Kind! **

On September 8, 2010 two small asteroids came within 80,000 km of Earth. Their small size of only 15 meters made
them very hard to see without telescopes pointed in exactly the right direction at the right time. In this problem, based on a
NASA press release,
students use a simple formula to calculate the brightness of these asteroids from their distance and size.
[Grade: 8-10 | Topics: Evaluating a base-10 power function; graphing; astronomical brightness scale]
(PDF)

**Problem 365: Meteorite Compositions: A matter of density **

Astronomers collect meteorites to study the formation of the solar system 4.5 billion years ago.
In this problem, students study the composition of a meteorite in terms of its density and mass,
and the percentage of iron and olivine to determine the volumes occupied by each ingredient.
[Grade: 8-10 | Topics: density; mass = density x volume; percentages]
(PDF)

**Problem 612: Exploring Power-laws: Meteor impacts **

Students work with logarithmic functions, power-laws and explore the mass functiuon of meteors.
[Grade: 9-12 | Topics: logarithmic functions; power-functions; logarithmic graphs]
(PDF)

**Problem 340: Computing the Orbit of a Comet**
Students use data from the orbit of Halley's Comet to determine the equation for its elliptical orbit.
[Grade: 10-12 | Topics: ellipses; solving quadratic systems; fitting ellipse to data points ]
(PDF)

**
Problem 338: Asteroids and Ice **
Students calculate how much ice may be present on the asteroid 24-Themis based on recent discoveries by NASA
[Grade: 9-12 | Topics: mass=densityxvolume; volume of a spherical shell]
(PDF)

**Problem 333: Hubble: Seeing a Dwarf Planet Clearly**
Based on a recent press release, students use the published photos to determine the sizes of the smallest discernible features and compare them to the sizes of the 48-states in the USA.
They also estimate the density of Pluto and compare this to densities of familiar substances to create a 'model' of Pluto's composition.
A supplementary Inquiry Problem asks students to model the interior in terms of two
components and estimate what fraction of Pluto is composed of rock or ice.
[Grade: 8-12 | Topics: scales and ratios; volume of sphere; density=mass/volume]
(PDF)

**
Problem 332: Hubble: The Changing Atmosphere of Pluto**
Based on a recent press release, students determine the aphelion and perihelion of Pluto's elliptical orbit using the properties of ellipses, then
calculate the temperature of Pluto at these distances to estimate the thickness of Pluto's atmosphere and its changes during its orbit around the sun.
[Grade: 10-12 | Topics: properties of ellipses; evaluating an algebraic function ]
(PDF)

**
Problem 331: Webb Space Telescope: Detecting dwarf planets**
The 'JWST' will be launched some time in 2014. One of its research goals will be to detect new dwarf planets beyond the orbit of Pluto.
In this problem, students use three functions to predict how far from the sun a body such as Pluto could be detected, by calculating
its temperature and the amount of infrared light it emits.
[Grade: 9-12 | Topics: Evaluating square-roots and base-e exponentials]
(PDF)

**
Problem 326: Hubble Spies Colliding Asteroids**
Based on a recent press release, students calculate how often asteroids collide in the Asteroid belt using a simple formula. Students
estimate belt volume, and asteroid speeds to determine the number of years between collisions. They also investigate how
the collision time depends on the particular assumptions they made. An 'extra' integration problem is also provided for calculus students.
[Grade: 8-12 | Topics: Volume of a thin disk; Algebra 1; Evaluating a definite integral; power-law]
(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]
(PDF)

**
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]
(PDF)

**
Problem 255: Tempel-1 - Closeup of a Comet **
Students examine an image of the Comet Tempel-1 taken by the Deep Impact spacecraft to determine feature sizes and other details.
[Grade: 6-8 | Topics: scales, proportions ]
(PDF)

** Problem 143: So..How big is it? - Asteroid Eros surface**
Students calculate the scale of an image of the surface of the asteroid Eros from the NEAR mission, and
determine how big rocks and boulders are on its surface.
[Grade: 4 - 7 | Topics: Scaling; multiplication, division; metric measure]
(PDF)

** Problem 116: The Comet Encke Tail Disruption Event **
On April 20, 2007 NASA's STEREO satellite captured a rare
impact between a comet and the fast-moving gas in a solar coronal mass ejection. In this problem, students
analyze a STEREO satellite image to determine the speed of the tail disruption event.
[Grade: 8-10 | Topics:time calculation; finding image scale; calculating speed from distance and time]
(PDF)

**Problem 57: Asteroids and comets and meteors - Oh My! ** -
Astronomers have determined the orbits for over 30,000 minor planets
in the solar system, with hundreds of new ones discovered every year. Working from a map of the locations
of these bodies within the orbit of Mars, students will calculate the scale of the map, and answer questions
about the distances between these objects, and the number that cross earth's orbit. A great, hands-on
introduction to asteroids in the inner solar system! Links to online data bases for further inquiry are also
provided.
[Grade level: 4-6 | Topics: Scale model; Decimal math; Interpreting 2-D graph]
(PDF)

**Problem 542:The Late Heavy Bombardment Era**

Students estimate the average arrival time of large asteroids that impacted the moon. They work with the formula for the volume of a sphere to estimate how much additional mass was added to the moon and Earth durung this era.
[Grade: 6-8 | Topics: volume of spheres; proportions]
(PDF)

**Problem 541:How to Build a Planet**

Students study planet growth by using a clay model of planetessimals combining to form a planet by investigating volume addition with spheres.
[Grade: 3-5 | Topics: graphing; counting]
(PDF)

**Problem 496: How to Grow a Planet or a Rain Drop **

Students use calculus to slove for the growth in mass of a body, and solve the equation for the case of a raindrop and a planet like Earth.
[Grade: 12 | Topics: Solving a simple differential equation.]
(PDF)

**Problem 464: Big Moons and Small Planets **

Students work with a scaled drawing of 26 large moons in the solar system, and together with an exercise in using simple
fractions, explore the relative sizes of the moons compared to Earth.
[Grade: 3-5 | Topics: scale models; proportions; fractions]
(PDF)

**
Problem 305: From Asteroids to Planetoids**
Students explore how long it takes to form a small planet from a collection of asteroids in a planet-forming disk
of matter orbiting a star.
[Grade: 11-12 | Topics: Integral calculus]
(PDF)

**
Problem 304: From Dust Balls to Asteroids**
Students calculate how long it takes to form an asteroid-sized body using a simple differential equation.
[Grade: 11-12 | Topics: Integral Calculus]
(PDF)

**
Problem 303: From Dust Grains to Dust Balls**
Students create a model of how dust grains grow to centimeter-sized dust balls as part of forming a planet.
[Grade: 11-12 | Topics: Integral Calculus]
(PDF)

**
Problem 302: How to Build a Planet from the Inside Out**
Students model a planet using a spherical core and shell with different densities. The goal is to create a planet of the
right size, and with the correct mass using common planet building materials.
[Grade: 9-11 | Topics: Geometry; volume; scientific notation; mass=density x volume]
(PDF)

**
Problem 249: Spotting an Approaching Asteriod or Comet **
Students work with a fundamental equation for determing the brightness of an asteroid from its size and distance from Earth.
[Grade: 10-12 | Topics: Algebra 1, logarithms, area, scientific notation]
(PDF)

** Problem 59: Getting A Round in the Solar System! ** -
How big does a body have to be before it becomes round? In this
activity, students examine images of asteroids and planetary moons to determine the critical
size for an object to become round under the action of its own gravitational field. Thanks to many Internet image archives
this activity can be expanded to include dozens of small bodies in the solar system to enlarge the
research data for this problem. Only a few example images are provided, but these are enough for
the student to get a rough answer!
[Grade level: 6-8 | Topics: Data analysis; decimals; ratios; graphing]
(PDF)