rich mathematical problems in astronomy

RICH MATHEMATICAL

PROBLEMS IN ASTRONOMY

Sandra Miller and Stephanie SmithLamar High School

Arlington, TX

HOW FAR AWAY IS THE HORIZON?

DISTANCE TO THE HORIZON

This problem is designed to occur during a Geometry unit on circles.

A line tangent to a circle forms a right angle with a radius drawn at the point of tangency.

DISTANCE TO THE HORIZON

r – radius of the planet/moon

h – height of the observer (eyes)

d – distance to the horizon

r

rh

d

DISTANCE TO THE HORIZON

r – radius of the planet/moon

h – height of the observer (eyes)

d – distance to the horizon

r

rh

d

2 2d r h r

2 2 22d r rh h r 2d h r h

DISTANCE TO THE HORIZON

Object Radius HorizonEarth 3959 mi. 3 mi.Moon 1080 mi.Mars 2106 mi.

Jupiter 43,441 mi.

DISTANCE TO THE HORIZON

Object Radius HorizonEarth 3959 mi. 3 mi.Moon 1080 mi. 1.6 mi.Mars 2106 mi. 2.2 mi.

Jupiter 43,441 mi. 9.9 mi.

MASS AND ESCAPE VELOCITY

MASS AND ESCAPE VELOCITY This problem set is geared toward a Pre-

AP Algebra I class or an Algebra II class.

By working through this packet, a student will practiceSimplifying literal equationsCreating formulasUnit conversionsUsing formulas to solve problems

MASS AND ESCAPE VELOCITYSir Isaac Newton developed three equations that we will use to develop some interesting information about the solar system.

When a force F acts on a body of mass m, it produces in it an acceleration a equal to the force divided by the mass.The centripetal acceleration a of any body moving in a circular orbit is equal to the square of its velocity v divided by the radius r of the orbit.The grativational force F between two objects is proportional to the product of their two masses, divided by the distance between them.

F ma

2va

r

1 22

GmmFr

MASS AND ESCAPE VELOCITY If we substitute the formula for

centripetal acceleration into the F = ma equation, we have an equation for the orbital force:

The gravitational force that the object being orbited exerts on its satellite is

2 2v mvF mr r

2GmMF

r

MASS AND ESCAPE VELOCITY Objects that are in orbit stay in orbit

because the force required to keep them there is equal to the gravitational force that the object being orbited exerts on its satellite.

If we set our two equations equal to each other and solve for v, we end up with a formula that will give us the orbital speed of the satellite.

MASS AND ESCAPE VELOCITY Simplify the equation and solve for v:

2

2mv GmM

r r

MASS AND ESCAPE VELOCITY Simplify the equation and solve for v:

2

2mv GmM

r r

2 GmMmvr

2 Gmvr

GMvr

MASS AND ESCAPE VELOCITY Because the mass of the satellite m

cancelled out of the equation, if we know the orbital velocity and the radius of the orbit, we can find the mass of the object being orbited.

MASS AND ESCAPE VELOCITY Rewrite the velocity equation and solve

for M:

2 GMvr

MASS AND ESCAPE VELOCITY Rewrite the velocity equation and solve

for M:

2 GMvr

2v r GM

2v rMG

MASS AND ESCAPE VELOCITY Example: Use the Moon to calculate the

mass of the Earth.

Orbital radius: Period: T = 27.3 days

Orbital velocity:

83.84 10 mr

circumference of orbit

period of orbitv

MASS AND ESCAPE VELOCITY Example: Use the Moon to calculate the

mass of the Earth.

2 rvT

82 3.84 1024 hours 3600 seconds27.3 1 day 1 hour

m1023 s

MASS AND ESCAPE VELOCITY Example: Use the Moon to calculate the

mass of the Earth.

2v rMG

2112m6.67 10 N kgG

246.02 10 kg

MASS AND ESCAPE VELOCITY To calculate escape velocity, we set the

equation for kinetic energy to the equation for gravitational force and solve for v:

Kinetic energy > Force × distance 22

1 2GmMmv r

r

2 2GMvr

2GMv

r

MASS AND ESCAPE VELOCITYCalculate Earth’s escape velocity in km/s.

Earth’s mass: 6.02 × 1024 kg Earth’s radius: 6.38 × 106 m

km11.22 sv

MASS AND ESCAPE VELOCITY Now that we’ve worked through the

different equations, we can calculate the mass and escape velocity of Mars as well as the mass of the Sun.

ASTRONOMY PROBLEMS

One of my favorite sites for possible astronomy-related math problems has been Space Math athttp://spacemath.gsfc.nasa.gov.

Unfortunately, because of cutbacks in NASA’s education budget, it will not be updated as frequently.

RICH MATHEMATICAL TASKS

Original (Standard) Problem

Invert the problem

Ask for prediction

Break into multiple

parts

Ask for multiple

representation

Ask questions

that require qualitative reasoning

Automaticity practice

Ask for generalizatio

n

Examples or counter-

examples

Ask for an explanation:

oral or written

James Epperson, Ph.D.

PRESENTATION MATERIALS The powerpoint and the worksheets will

be posted on my blog at tothemathlimit.wordpress.com.