PRIOR KNOWLEDGE
1. Constant Speeds Around a Circle
Jose and Sue run laps on a circular track. Jose runs in a lane that measures 270 meters per lap and Sue
runs in a lane that measures 180 meters per lap. Both run at uniform speeds with Jose completing each
of his laps in 90 seconds. After 6 minutes, Sue has run exactly one more lap than Jose.
a. How long does it take Sue to complete each lap?
b. How many “degrees” does Sue gain on Jose every time she completes a lap?
c. Who is running the faster?
ANSWER: Jose completes 4 laps in 6 minutes (360 seconds), whereas Sue completes 5 laps; Sue requires
72 seconds per lap. Jose’s speed is 270 m/90 sec = 3 m/sec whereas Sue’s speed is 180 m/72 sec = 2.5
m/sec. Sue runs at the rate of 5o per second, whereas Jose runs at 4o per second; during the time for Sun
to complete a lap (72 sec), Jose complete (4)(72)=288o. Sue gains 360o-288o=72o; during her 5 laps, she
gains 360o, or one complete lap.
2. Eclipse and Lunar Phases
a. Indicate the relative positions of the earth, moon, and sun (need not be drawn to scale)
o when the moon is in its ”full” phase and when it is in its “new” phase.
o when the moon is in its “new” phase.
o during a lunar eclipse.
o during a solar eclipse.
b. Explain why lunar and solar eclipses do NOT occur every month.
ANSWERS. The configuration during the full moon and lunar eclipse will look the same; likewise
the configurations of the new moon and solar eclipse look similar. The plane of the orbit of the
moon around the earth is not the same as the plane of the orbit of the earth around the sun. As a
result, the moon may be above or below the earth’s shadow during a full moon and the earth may
be above or below the moon’s shadow during a new moon.
3. Reason for the Seasons.
a. Explain why summer is generally warmer than winter.
b. Indicate evidence that refutes the reason is due primarily to the proximity of the earth to the sun.
o The shadows are longer in the winter.
o The sun rises earlier and sets later in the summer.
o The sun appears higher in the sky in June than in December.
4. GeoGebra’s Angle Measure. The figure below indicates how GeoGebra measures angles: angles are
measured counter-clockwise from the first segment to the second segment. In the example, for angle
BCA, the first segment is BC and the second segment is CA.
Exercises.
a. Indicate the measure of angles FDE & EFD (as indicated above)
b. Given the points G & H, determine a point X such that GHX=300o and HGX=60o.
VOYAGE TO A DISTANT PLANET
5. The First Observations from a New Planet
In the year 2222, the United States launched a spacecraft on a voyage to the SUN, a nearby star. The
time for this trip was shortened by using a worm hole. After several (earth) months they arrived at a
previously undiscovered planet which they named Planet B. The astronauts found that the pressure,
temperature, and atmospheric constituents comparable to what they experienced on earth. They
noticed that the SUN resembled their familiar sun. There were two other planets which they named
Planets A and C, plus a MOON. After a few days they began their observations. They set up a twometer pole and observed the shadow due to the SUN. The minimum length of the shadow remained
at exactly one meter day after day.
QUESTION: WHAT COULD THEY INFER?
ANSWER: The planet was NOT tilted on its axis and there are no “seasons”.
6. Observations of the SUN and MOON.
The MOON had its lit surface appear to change, going through a complete cycle every 20 days. Every
cycle there was a short period of time in which the MOON exactly covered the SUN, and exactly 10
days later darkness suddenly fell on the MOON for several hours. Whenever they held a small
button (diameter=0.9 cm) at a distance of 60 centimeters from their right eyes, the coin exactly
covered up the MOON. WHAT COULD THEY INFER?
ANSWERS:
o The plane of the MOON was the same as the plane of the planet around the SUN.
o The orbit of the MOON is circular around the planet (since its distance from the planet does
not change).
o If rS is the radius of the SUN, dS is its distance of D from the planet, rM is the radius of the
MOON and dM is the distance of the MOON from the planet, then rS/dS=rM/dM=.9/60=.015 .
o The duration of the the MOON eclipse can be estimated. The shadow moves a distance 2rM
and the speed is the circumference of the orbit 2πdM divided by 200 hours. The duration is
approximately 8/π ~2.5 hours.
7. Observations of Planets A and C.
 Planet B requires 270 days to complete one revolution around SUN.
 If Planet A is directly between Planet B and the SUN on a given day, the next time Planet A is
directly between Planet B and the SUN occurs 135 days later.
 If Planet B is directly between Planet C and the SUN on a given day, the next time Planet B is
directly between Planet C and the SUN is 405 days later.
QUESTIONS:
a. At what rate is Planet B moving around the sun (in degrees per day)?
ANSWER: 360/270=4/3 deg/day.
b. At what rate is Planet A moving ahead of Planet B? ANSWER: 360/135=8/3 deg/day.
c. At what rate is Planet A moving around the SUN (in degrees per day)?
ANSWER: (4/3)+(8/3)=4 deg/day
d. How long in years does it take Planet A to complete one orbit around the SUN?
ANSWER: 360/4=90 days=(1/3) year
e. How long in years does it take Planet C to complete one orbit around the SUN?
ANSWER:
 Planet C falls behind Planet B at the rate of 360/405=8/9 deg/day.
 Planet B is moving at the rate 4/3 deg/day.
 Planet C must be moving around the SUN at the rate (4/3)-(8/9) =4/9 deg/day
 Planet C requires 360/ (4/9) = 810 days or 810/270= 3 years to complete one revolution.
8. Determination of the Distance of Planet A from the SUN.
Thirty days after Planet A is between and aligned with the SUN and Planet B, Planets A and B
have moved a predictable amount. The scientists then make an observation: the angle PlanetAPlanetB-SUN measures 27.3o .
a. Explain why Planet B has moved 40 degrees around the SUN.
b. Explain why Planet A has moved 80 degrees ahead of Planet B.
c. Explain how we can use this information to determine the location of Planet A after 30 days.
What assumptions did you need to make in order to make this determination?
d. Use the figure below (left) as a template to construct an accurate sketch with a compass, ruler
and proctractor to estimate the distance of Planet A from the SUN in units of the distance
between SUN and Planet B.
DAY
Angle A-B-SUN (degrees)
Angle Sun-B-C (degrees)
0
0
180.0
30
27.3
132.6
60
6.5
98.3
90
341.4
72.7
e. Is your result consistent with Kepler’s third law, i.e.,
[distance from the sun]3 = [period around the sun]2 ?
ANSWER: The result is consistent with Kepler’s law.
f. Perform a construction to predict the measure of angle A-B-SUN after 60 days, then compare
with the result in the table.
9. How large is Planet B?
The astronauts now attempt to estimate the size of Planet B. They use their two-meter measuring stick
and measure the smallest length of its shadow during daylight hours, obtaining 1.0 meters. A small
group then traveled 500 kilometers due north and repeated the measurement, this time the minimum
shadow length had increased to 1.2 meters.
a. Use the GeoGebra sketch “size of planet” (figure above right) to simulate both situations by
adjusting the slider for the shadow length and the latitude.
b. What fraction of the circumference of the planet’s circumference had they traveled?
c. Estimate the diameter of Planet B.
ANSWERS:
 Adjust the latitude so that the dashed line is parallel to the rays of the sun. The initial
location is 27o latitude and the second location is around 31o latitude. The two locations are
4/360=1/90 of the planet’s circumference.
 The circumference is 90*500km=45000 km; therefore the planet’s diameter is 45000 km/π ~
14300 km.