Space Math

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Space Math                           

 

    http://spacemath.gsfc.nasa.gov 

 

Playing Baseball on the Earth-like Planet Kepler-22b! 

31 

 

NASA's Kepler mission has confirmed 

its first planet in the "habitable zone," the 

region where liquid water could exist on a 

planet’s surface.  

 

 

The newly confirmed planet, Kepler-

22b, is the smallest yet found to orbit in the 

middle of the habitable zone of a star similar 

to our sun.  

 

 

The planet is about 2.4 times the 

radius of Earth. Scientists don't yet know if 

Kepler-22b has a rocky, gaseous or liquid 

composition, but its discovery is a step closer 

to finding Earth-like planets. 

 

Problem 1 - Suppose Kepler-22b is a spherical, rocky planet like Earth with an average 

density similar to Earth (about 5,500 kg/meter

3

). If the radius of Kepler-22b is 15,000 km,  

what is the mass of Kepler-22b in A) kilograms? B) multiples of Earth’s mass (5.97x1024 

kg)?  

 

Problem 2 - The acceleration of gravity on a planetary surface is given by the formula 

 

                                                 

2

GM

a

R

=

   

 

where M is in kilograms, R is in meters and G is the Newtonian Constant of Gravity with a 

value of 6.67 x 10

-11

 m

3 

kg

-1

 sec

-2

   What is the surface acceleration of Kepler-22b A) In 

meters/sec

2

? B) In multiples of Earth’s surface gravity 9.8 meters/sec

2

? 

 

Problem 3 - The relationship between surface acceleration and your weight is a direct 

proportion. The surface acceleration of Earth is 9.8 meters/sec

2

. If you weigh 150 pounds 

on the surface of Earth, how much will you weigh on the surface of Kepler-22b? 

 

Problem 4 -  The dimensions of a typical baseball park are determined by the farthest 

distance that an average batter can bat a home-run. This in turn depends on the 

acceleration of gravity, which is the force that pulls the ball back to the ground to shorten 

its travel distance. For a standard baseball field, the distance to the back-field fence from 

Home Plate may not be less than 325 feet, and the baseball diamond must be exactly 90 

feet on a side. 

 

A) If the maximum travel distance of the baseball scales linearly with the acceleration of 

gravity, what is the minimum distance to the back-field fence from Home Plate along one 

of the two foul lines? 

 

B) What are the dimensions of the baseball diamond? 

 

An artists illustration of Kepler-22b 

(Credit NASA/Ames/JPL-Caltech) 

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Space Math                           

 

    http://spacemath.gsfc.nasa.gov 

 

 

Problem 1 - Suppose Kepler-22b is a spherical, rocky planet like Earth with an 

average density similar to Earth (about 5,500 kg/meter

3

). If the radius of Kepler-22b is 

15,000 km, what is the mass of Kepler-22b in A) kilograms? B) multiples of Earth’s 

mass (5.97x10

24

 kg)? 

 

Answer: A) First find the volume of the spherical planet in cubic meters, then multiply 

by the density of the planet to get the total mass. 

                    R = 15,000 km x (1000 m/1 km) = 1.5 x 10

7

 meters. 

 

V = 4/3 

π

 R

3

 

    = 1.33 x 3.14 x (1.5 x 10

7

 meters)

3

     =  1.41x10

22

 meters

3

 

 

Then M =  density x volume 

             =  5,500 kg/m

3

 x (1.41x10

22

) 

             =  7.75 x10

25

 kg 

 

B)  M = 7.75x10

25

 kg / 5.97x10

24

 kg   = 12.9 Earths. 

 

 

Problem 2 -  A) In meters/sec

2

? B) In multiples of Earth’s surface gravity 9.8 

meters/sec

2

? 

 

Answer:  A)  a = 6.67 x 10

-11

 (7.75x10

25

)/

 (

1.5 x 10

7

 )

2

  = 23.0 meters/sec

2

 

                B)  23.0 / 9.8 =  2.3 times earth’s surface gravity 

 

Note: From the formula for M and a, we see that the acceleration varies directly 

with the radius change, which is a factor of 2.4 times Earth, so a = 2.4xa(earth) 

 

Problem 3 – The relationship between surface acceleration and your weight is a direct 

proportion. The surface acceleration of Earth is 9.8 meters/sec

2

. If you weigh 150 

pounds on the surface of Earth, how much will you weigh on the surface of Kepler-

22b? 

 

Answer:  By a simple proportion:  X/150 = 2.3/1.0  so  x = 2.3x150 = 345 pounds. 

 

 

Problem 4 -  For a standard baseball field, the distance to the back-field fence from 

Home Plate may not be less than 325 feet, and the baseball diamond must be exactly 

90 feet on a side. 

 

A)  If the maximum travel distance of the baseball scales linearly with the 

acceleration of gravity, what is the distance to the back-field fence from Home 

Plate along one of the two foul lines?  Answer:   325 / 2.3 = 141 feet. 

 

B)  What are the dimensions of the baseball diamond? Answer: 90/2.3 = 39 feet on 

a side. 

Answer Key 

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