Adventures in Math
Author, Dr. Sten Odenwald
The goal of these space problems is to teach students about astronomy and space science by using mathematics and realworld problems. Included with each problem is an answer key and a note to the teacher. Check out the problems relevant to the Deep Impact mission below.

Figure showing flyby geometry. Credit: NASA/Space Math 
Deep ImpactComet Flyby
On July 4, 2005, the Deep Impact spacecraft flew by the comet Tempel 1
along a path at a speed of 10 km/sec. Its closest distance to the comet was
b = 500 kilometers at a time, t=0. The distance traveled along the path is given by l = Vt.
What is the formula for the distance to the comet from the spacecraft?
+ Solve this problem and others
Tempel 1  Closeup of a Comet!
On June 4, 2004 the Deep Impact spacecraft flew within 500 km of comet Tempel 1 and created an image of its surface, before it began to eject huge plumes of gas to form its tail.
What is the size of some of the smallest details you can see on its surface?
+ Solve this and other problems

This image shows initial ejecta that resulted when NASA's Deep Impact probe collided with comet Tempel 1. Credit:NASA/Space Math 
Deep Impact Comet Encounter
On July 5, 2005 at 5:45 UT the 362kilogram Impactor from NASA's Deep Impact mission, collided with the nucleus of the comet Tempel 1, causing a bright flash of light and a plume of ejected gas. Traveling at nearly 10.3 km/sec, the Impactor created a crater on the nucleus and ejected about 10,000 toms of material. The average density of the comet nucleus is 400 kg/m^{3} and its size can be approximated as a sphere with a radius of 3 kilometers.
From the information given, what was the approximate mass of the comet nucleus in kilograms?
+ Solve this and other problems

"Skinny triangle" depicting relationship of diameters of Earth and Moon, with ratio dependent on how far observer is when he sees them. Credit: NASA/Space Math 
Earth and Moon Angular Sizes
In space, your perspective can change in complicated ways that sometimes go against Common Sense unless you 'do the math'. This happens very commonly when we are looking at one object pass across the face of another. Even though the Moon is 1/4 the diameter of Earth, the simple ratio of their apparent diameters will depend on how far from them YOU are when you see them. In the figure above, assume that the diameter of the Moon is less than 1 degree when spotted by the spacecraft located at 'O'.
What is the angular diameter of the Moon, θsubm, in terms of d and R?
+ Solve this problem and others
For more space math, go to http://spacemath.gsfc.nasa.gov