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Space: Answers
We Have Liftoff
| Orbiting The Earth
| Solar System
| Answers
| More Math Concepts
We
Have Liftoff: Math and the Space Program
1.
Take 363 divided by 82.5, 90, and 184.2 for the Mercury,
Gemini, and Space Shuttle launches, respectively.
The Apollo/Saturn V vehicle was 4.4 times taller than
the Mercury/Atlas vehicle.
The Apollo/Saturn V vehicle was 4.0 times taller than
the Gemini/Titan II vehicle.
The Apollo/Saturn V vehicle is 2 times taller than the
Space Shuttle Discovery.
2.
a) The radius of the capsule equals (6 + (2.5/12))/2
= 3.10 ft. The height of the capsule equals 6 + (10/12)
= 6.8 ft. The volume of the Mercury capsule equals 1/3
x (pi) x (3.1)2 x (6.2) = 68.4 cubic feet.
2.
b) The radius of the capsule equals 10/2 = 5 ft. The
height of the capsule equals 7 + (6/12) = 7.5 ft. The
volume of the Gemini capsule equals 1/3 x (pi) x (5)2
x (7.5) = 196.3 cubic ft. 196.3 cubic ft / 2 = 98.2
cubic ft per person.
2.
c) The radius of the capsule equals (12 + 6/12)/2 =
6.4 ft. The height of the capsule equals 10 + (7/12)
= 10.6 ft. The volume of the Gemini capsule equals 1/3
x (pi) x (6.25)2 x (10.6) = 454.7 cubic ft.
433.6 cubic ft/3 = 151.6 cubic ft per person.
3.
2625 cubic feet/7 people = 375 cubic ft per person.
4.
The Space Shuttle mission in hours equals 8 x 24 + 22
+ 4/60 = 214.07 hours. The Mercury mission in hours
equals 4 + 55/60 + (23/60)/60 = 4.92 hours. The Space
Shuttle mission is 214.07/4.92= 43.5 times longer.
Orbiting the Earth
1.
4 + 24/60)/3 = 1.47 hours or 88 minutes.
2.
4/24 x 88 minutes = 14.67 minutes.
3.
2,700 miles/17,544 mph = 0.15 hours or 0.15 x 60 = 9
minutes.
4.
To get the time in terms of hours add 4 + (24/60)) to
get 4.4 hours. 17,544 miles/hour x 4.4 hours = 77,193
miles. John Glenn traveled approximately 77,193 miles.
5.
77193 miles/3 orbits = 25,731 miles/orbit. 25,731 represents
the circumference of his orbit. Since circumference
= 2 x (pi) x radius, the radius of his orbit = 25,731/(2
x (pi)) = 4095 miles. 4095 represents the distance from
the center of the earth. His distance above the earth
would be 4095 miles - radius of the earth = 4095 miles
- (7928/2) miles = 131 miles. The actual elliptical
orbit ranged from 99 miles to 165 miles.
6.
The semi major axis a = (265 + 159)/2 + 6382
= 6594 kilometers. Period = 2 (pi) x square root (65943/398601)
= 5328 seconds = 88.8 minutes.
7.
The semi major axis a = 482 + 6382 = 6864 kilometers.
Period = 2 (pi) x square root (68643/398601)
= 5659 seconds = 94.3 minutes.
8.
The period is 24 hours = 24 x 60 x 60 = 86400 seconds.
Plugging this into the equation gives 86400 = 2 (pi)
x square root(a3/398601). Solving for a
gives you a = cube root ((86400/(2 x pi))2
x 398601) = 42241 kilometers. So the altitude above
earth = 42241 - 6382 = 35,859 kilometers above earth.
35,859 kilometers x 0.6215 = 22,286 miles above earth.
Modeling the Solar System
1.
Students may use student
data sheet 1. Answers are in column 3 of the table
of Size
Data for the Planets in the Solar System. Answers
will vary slightly depending on when you round.
2.
Answers will vary. If students pick the earth to be
the size of a tennis ball, with a diameter of about
2.5 inches, then Jupiter (the largest planet) would
have a diameter of 28 inches, and Pluto (the smallest
planet) would have a diameter of about half an inch.
If students pick the earth to be the size of a basketball,
with a diameter of about 9.5 inches, then Jupiter would
have a diameter of 106 inches, and Pluto would have
a diameter of about 1 3/4 inches.
3.
Possible answers using a tennis ball or basketball are
in columns 4 & 5 of the table of Size
Data for the Planets in the Solar System. Answers
will vary depending on the sizes that students choose.
4.
Answers will vary. If students pick the earth to be
the size of a tennis ball, with a diameter of about
2.5 inches, then the Sun would have a diameter of 272
inches (almost 23 feet or the height of a two story
building). If students pick the earth to be the size
of a basketball, with a diameter of about 9.5 inches,
the Sun would have a diameter of 1035 inches (86 feet
or the height of an eight story building).
5.
Students may use student
data sheet 2. Answers are in column 3 of the table
of Distance
from Sun Data for Planets in Our Solar System. Answers
will vary slightly depending on when you round.
6.
Students may use student
data sheet 3. Answers will vary depending on the
size of earth chosen. Sample answers are in the table
for Earth using the size of a tennis ball and a basketball;
they will vary slightly depending on when you round.
7.
Students may continue to use student
data sheet 3. Answers will vary. You may want to
divide the answers in inches by 12 to convert to feet.
For some values, you may want to divide the feet by
5280 to get miles. See the table
for Earth using the size of a tennis ball and a basketball;
they will vary slightly depending on when you round.
8.
Using the tennis ball or basketball for Earth will not
allow the models to fit on a football field. In the
case of the tennis ball for Earth, Pluto is about 18.29
miles from the Sun. In the case of the basketball for
Earth, Pluto is about 69.5 miles from the Sun. Depending
on the size of your city, these may or may not fit within
your city.
9.
Answers will vary depending on scale and location of
your school. This is a good way for students to start
to get a feel for the size and distance between planets.
Find a convenient place to use as the location of the
Sun, and pace off distances for the near planets. You
can also use a map to plot where in your city or state
the other planets will be.
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