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Analyzing Planets' Gravity and Periods

Activity Description Activity GuideResources


Planet Period 
(in days)
Distance from Sun (106k)
Gravity 
(in m/sec2)
Mercury  88   57.900  3.70
Venus  225   108.200  8.87
Earth  365   149.600  9.78
Mars  687   227.900  3.69
Jupiter  4332   778.300  23.12
Saturn  10760   1427.000  8.96
Uranus  30685   2869.328  8.69
Neptune  60189   4496.672  11.00
Pluto  90456   5913.500  0.66

 

Part 1: Analyzing and Visualizing Freefall

(1) Write an equation for the downward distance traveled by an object in freefall as a function of time.
Note: The general equation for vertical projectile motion is h(t) = ho + vt - .5gt2, where g is the gravitational constant.

(2) Write an equation to describe the height, at time t, of an object free falling under the influence of gravity from 500 meters above the surface of the Earth.

(3) Write similar equations for the other planets.

(4) Discuss the differences between the downward accelerations of these objects. Predict how long it would take objects to hit the surfaces of these planets. Also, predict how long it will take each object to fall halfway to the planet's surface

(5) Use parametric equations to simulate on your graphing calculator the motion of objects free falling from 500 meters above each of the planets, using parametric graphs in plot mode. Label your graphs, if practical.

Suggestion: Fix the planets' downward graphs at x=1, x=2, etc. So, for example, the parametric equations for Mercury are x(t)=1, y=500-.5(3.7)t2.

Note: Some graphing calculators can only graph 6 parametric equations simultaneously.

(6) Trace each planet's graph and observe the times it takes for the objects to hit the surface of each planet. Compare the falling times to those predicted in task (4). Reconcile your predictions with your observations. Discuss your findings.

(7) Approximate the values of the derivative at several points along the downward path of the object falling towards the Earth. Use your calculators' numerical derivative capabilities to check your predictions. Reconcile the results.

(8) When tracing the path of an object falling towards Earth, note the difference in the distance fallen each second. Relate your observations to the gravity constant, acceleration, and the second derivative.

(9) Discuss advantages and disadvantages of using plot mode for such graphs.

Part 2: Deriving Kepler's Third Law:

(10) Examine the periods and the average distances from the sun for the different planets, and describe any relationship(s) you see between these variables.

(11) Enter this data into lists in your calculator and draw a scatterplot. Discuss how this scatterplot fits with the relationships you described in Task 7.

(12) Discuss viable regression curves to best fit the data. Explain your choices.

(13) Use your calculator to calculate several regression models and decide on a "best fit" model.

(14) The power regression fits the data with r = .99999. Use the power model to write an expression for distance from the sun distance as a function of period [d=(2.9)p.66 ], and then rewrite it without fractional exponents. Put this relationship into natural language.

(15) Use any resource you can to find a statement of Kepler's Third Law. Compare this law with your best-fit relationship.

Note: You may find Kepler's Third Law stated in two slightly different ways:
(1) The square of the period (year) of a planet is proportional to the cube of the major axis of the orbit. (2) The square of the period of a planet is proportional to the cube of a planet's mean distance from the sun.


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Last modified on August 15, 2001.