rich mathematical problems in astronomy

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RICH MATHEMATICAL PROBLEMS IN ASTRONOMY Sandra Miller and Stephanie Smith Lamar High School Arlington, TX

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Rich Mathematical Problems in Astronomy. Sandra Miller and Stephanie Smith Lamar 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. - PowerPoint PPT Presentation

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Page 1: Rich Mathematical Problems in Astronomy

RICH MATHEMATICAL

PROBLEMS IN ASTRONOMY

Sandra Miller and Stephanie SmithLamar High School

Arlington, TX

Page 2: Rich Mathematical Problems in Astronomy

HOW FAR AWAY IS THE HORIZON?

Page 3: Rich Mathematical Problems in Astronomy

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.

Page 4: Rich Mathematical Problems in Astronomy

DISTANCE TO THE HORIZON

r – radius of the planet/moon

h – height of the observer (eyes)

d – distance to the horizon

r

rh

d

Page 5: Rich Mathematical Problems in Astronomy

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

Page 6: Rich Mathematical Problems in Astronomy

DISTANCE TO THE HORIZON

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

Jupiter 43,441 mi.

Page 7: Rich Mathematical Problems in Astronomy

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.

Page 8: Rich Mathematical Problems in Astronomy

MASS AND ESCAPE VELOCITY

Page 9: Rich Mathematical Problems in Astronomy

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

Page 10: Rich Mathematical Problems in Astronomy

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

Page 11: Rich Mathematical Problems in Astronomy

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

Page 12: Rich Mathematical Problems in Astronomy

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.

Page 13: Rich Mathematical Problems in Astronomy

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

2

2mv GmM

r r

Page 14: Rich Mathematical Problems in Astronomy

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

2

2mv GmM

r r

2 GmMmvr

2 Gmvr

GMvr

Page 15: Rich Mathematical Problems in Astronomy

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.

Page 16: Rich Mathematical Problems in Astronomy

MASS AND ESCAPE VELOCITY Rewrite the velocity equation and solve

for M:

2 GMvr

Page 17: Rich Mathematical Problems in Astronomy

MASS AND ESCAPE VELOCITY Rewrite the velocity equation and solve

for M:

2 GMvr

2v r GM

2v rMG

Page 18: Rich Mathematical Problems in Astronomy

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

Page 19: Rich Mathematical Problems in Astronomy

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

Page 20: Rich Mathematical Problems in Astronomy

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

Page 21: Rich Mathematical Problems in Astronomy

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

Page 22: Rich Mathematical Problems in Astronomy

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

Page 23: Rich Mathematical Problems in Astronomy

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.

Page 24: Rich Mathematical Problems in Astronomy

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.

Page 25: Rich Mathematical Problems in Astronomy

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.

Page 26: Rich Mathematical Problems in Astronomy

PRESENTATION MATERIALS The powerpoint and the worksheets will

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