# Gravity - What happens when two objects of unequal masses fall freely towards the ground? (Revisited)

The common perception regarding what happens when two objects of equal sizes but unequal mass are allowed to fall freely towards the ground is that - both the objects make contact with the ground at the same instant.

This is attributed to the fact that the acceleration of both the objects towards the earth are the same for a given height. This was the reason for Galileo's expirement.

But the above fact ignores the fact that each of the three objects are attracting each other. In particular, the heavier object also attracts the lighter object, thus decreasing the distance between the lighter object and earth.

Note : I am ignoring the fact that the lighter object would also attract the heavier object (but to a lesser extent) and since the earth moves approximately in the same direction as a result of the attraction from the other two spheres, that is also ignored. Otherwise the earth can be said to move diagonally towards the heavier object (again to a small extent).

I have included an illustration showing the movement of the lighter object towards the heavier one.

As such the lighter object should fall first is it not?

Note : Generally, the two objects are placed close to each other (the distance-d between the objects is lesser than the height-h from the ground - 'd < h' ).

Continuing with the same logic, I have the following understanding :

When two spheres of equal size but unequal masses are allowed to fall freely towards the ground -

1.) From a height greater than the distance between the two spheres, the lighter sphere makes contact with the ground first.

2.) From same height as the distance between the two spheres, both the spheres make contact with the ground at the same time.

3.) From a height lesser than the distance between the two spheres, the heavier sphere makes contact with the ground first.

Note :

a.) The ground is also a sphere.

b.) The distance between the spheres are measured from their centers.

Is my understanding correct?

I also found this Phys.SE post. Which speaks of a scenario where in only two objects are involved. My question is similar to it.

PS: I have blogged about it before as well, saying it is so and also elaborated on two other cases ($d=h$ and $d>h$).

• Consider that the mass of the Earth is much much greater than the mass of the other objects is likely to be. This ensures that any movement of the Earth is negligible in comparison to all other distances involved. If that were not the case (all 3 masses were comparable) then you see that each body moved towards the center of mass of the other two. Take a look at the Three Body Problem, maybe? – Pranav Hosangadi Oct 29 '13 at 16:25
• @PranavHosangadi : What I am trying to say is- regarding the point of impact, the three conditions above hold good for all combinations of masses, not just because the earth is massive. The three objects are moving towards the center of gravity of the system. The system in this case involves the three objects. In case of the earth and two objects, the earth's mass implies that the center of gravity of the system is close to that of the earth, resulting in the earth moving a negligible distance. Thank you for the link. – Ravindra HV Oct 30 '13 at 7:16
• In practice, assuming both objects have the same shape and density distribution, and dropped from the same orientation, the heavier object should hit the ground first. – Cruncher Nov 22 '13 at 18:57
• What do you mean by 'orientation'? My understanding is that the heavier one will make contact only if the height (h) is lesser than the distance (d) (h<d). If h=d then both the objects make contact at the same instant. If h > d, then the lighter one would make contact first (the heavier object attracts the lighter one laterally thus decreasing the distance of the lighter object from the vertical, w.r.t the illustration above). I have blogged regarding it in detail. Search for "Effect of gravitational force upon three objects of unequal mass". It begins with a warning. – Ravindra HV Nov 24 '13 at 6:48
• the effect is very small because if M3 is more attracted by M2 than M1, this also brings M3 closer to M1 . – user46925 Feb 20 '16 at 2:19

It helps if you consider the components of the acceleration of the smaller planets due to the gravitation force of each other planet. Here is a rough diagram showing the components of acceleration for each planet, assuming the largest does not accelerate due to its large mass:

The red arrow shows the component of acceleration of a planet due to the gravity of the largest planet. The green arrow is the component due to the other planet (not the largest).

Now, consider the bodies accelerate only by the red components of acceleration (i.e. ignore the gravitation effects between the two smaller planets). As the centres of gravity for the smaller planets are the same distance away, the motion of the two smaller planets will be perfectly symmetrical, and both planets hit the largest at the same time (assume smaller planets have similar radii/size).

Now, let us add the effects of the green arrows (i.e. the gravitation effect between the smaller planets). The angle between the red arrow and the green arrow is less than 90°, so this means that the green arrow will add to the acceleration of the planet towards to largest planet. Let us assume that the green arrows are sufficiently small so that the total component of acceleration perpendicular to the line connecting each small planet to the largest planet does not cause large rotational effects about the largest planet (i.e. effects such that the small planets do not travel in approx. straight lines towards the largest planet, complicating matters, and possibly throwing the smaller planets into orbit around the larger!).

The less massive planet will have a larger green arrow, and the larger of the two green arrows has a greater contribution to the component of acceleration in the radial direction (radial, as in the line connecting the smaller planet to the biggest planet). This means that, in this instance, the less massive planet is accelerating slightly faster than the slightly more massive planet towards the biggest planet. From this, it is sufficient to assume that... drumroll the smaller planet will hit the largest planet first!

This is true for all values of d and h, assuming the values don't cause the smaller planets to collide with each other first, and that the path between each smaller planet, and the largest planet is approximately straight. (i.e. d is above a critical value)

• are you explaining on your own or elaborating my #1 point above ? – Ravindra HV Apr 2 '15 at 19:21

The gravitational forces acting on the two objects due to their interaction when 5 m apart have magnitudes of approx. 2.7e-2 N, giving initial horizontal accelerations of approximately 2.7e-8 and 2.7e-6 m/s^2. In the 1.49 s it takes for the two objects to reach Earth, those accelerations will change negligibly. The smaller mass will travel along a very slightly longer path but will have a very slightly higher average speed - why should the horizontal acceleration affect the vertical acceleration?

At a latitude of 45 degrees, the value of g varies by approximately 0.001 m/s^2 per degree latitude, which means that, if one mass is dropped from a point 5 m further north at that latitude, then its value of g will be roughly 4.5e-8 m/s^2 lower than the value for the other one, and it will hit the ground very slightly after the other one.

That's if one doesn't hit a can of worms.

• Sorry, not sure if I can follow you. But basically the idea is that while it 'appears' to be horizontal, with regard to the center-of-gravity, it contributes to the acceleration towards the center. – Ravindra HV Nov 18 '16 at 21:41

Two objects of unequal masses, ignoring other forces would be attracted by at the same velocity provided by the acceleration due to gravity. But objects with less mass are generally less dense, causing other agents to friction it's movement to the ground, while object which are more dense and have more mass will be more rigid to that force. This would cause the object with more mass to reach the ground faster than that which has less mass

• If I understood you correctly, you said that - a. its true that objects are attracted at the same rate independent of masses if we ignore forces b. you said that objects with more mass will be rigid - do you mean inertia ? c. but then you also said the object with more mass will reach ground faster, which (which contradicts b.) What I am trying to say above however is that - what falls first 'depends' on whether the 'angle' made by the falling objects to the reference object is acute, equilateral or obtuse – Ravindra HV Mar 28 '15 at 15:57

Only if the situation is symmetric and the masses are aligned like an equilateral triangle all masses will meet each other at the same time:

Code: .nb, .cdf

• The scenario I considered was where the three bodies were at rest without the influence of any external force and all the three were on the same plane. In your case, what you mentioned is the same as what was mentioned above. However, in the second animation, in case of the above one, two of the objects make contact before the third one comes in - which is similar to what I said in case of an isosceles triangle but they are not on the same plane. In case of the last animation, none of the three appear to start from rest. Which is also a different scenario from what I am considering. – Ravindra HV May 14 '16 at 14:54
• When I wrote the questions, I was not thinking of elastic collisions like you have. I only focused on the first point of contact. But I am still not sure if the collisions will result in circular motion. If they all start from rest, their paths should be linear is it not ? If you think about it, any three objects interacting with each other with no external influence will follow a path as if they are on the same plane. There is no scope for circular motion. They can never gain more energy than what they initially had. – Ravindra HV May 14 '16 at 15:16
• The center of gravity between the three masses cannot shift from what it was before. Their paths are initially linear and as per my understanding even after collision they cannot acquire a path that will either take them back to the initial state (and start all over again) or set them up again for yet another collision. In any case, the animations are interesting. Thanks! – Ravindra HV May 14 '16 at 15:16
• The scecond and third animation show the same scenario, the difference is only the trail length. They do not show any collision, neither elastic nor inelastic. And they are on the same plane, z=0 for all bodies. – Yukterez May 14 '16 at 17:25
• Ok. It makes sense now. Thanks. But the red and blue points do seem to make contact and deviate after that. That is what i meant by collision. Also towards the end of the animation, the center of gravity between the three objects appears to have shifted. Is it not ? – Ravindra HV May 15 '16 at 21:30

Well the purpose of the statement "both the objects make contact with the ground at the same instant" is to convey the fact that the gravitational acceleration does not depend upon mass of the falling object.

The experiment could have been done by dropping two bodies at different times and measuring the time they took to reach the ground and that would have given same results.

What you are pointing towards is - negligibles. Yes, there can be negligible effects due to simultaneous fall of two bodies but that does not change the message that was supposed to be conveyed.

• Perhaps, the intention was indeed to state it in spirit rather than in literal terms. However, it would still help to note it down somewhere, just in case there is a scenario where the casually stated sentence could prove costly. – Ravindra HV Feb 18 '17 at 13:41
• @RavindraHV: The angles you have shown are totally out of proportion, just to over highlight the effect. In practical, the lines are nearly parallel and the effect would not even be measurable to cause any damage. – kpv Feb 18 '17 at 16:34
• While I cannot indeed provide an example, still it does not imply equivalence. It's always worth it to keep the detains in scope. – Ravindra HV Feb 18 '17 at 17:45

If the mass of both the objects are high such that air resistance in negligible in them then both of them will touch the ground at the same time. If the body with smaller mass is so small that air resistance can't be neglected then the body with heavier mass will reach the ground faster.

• The point I am trying to make, is that although the earth attracts all objects at the same rate, the heavier objects attract the earth itself towards themselves as well, more so than the lighter objects. If the two other objects are far apart then its not easy to tell. However if they are far apart, in the best case at opposite ends, then its easier to tell. If the sun were to be shrunk to the size of the moon, and then the sun, earth and moon were placed at equal distances from each other, then the sun would definitely pull the earth towards itself far more than the earth would pull the sun! – Ravindra HV Feb 15 '15 at 19:05

Here is the output from a program I wrote - it analyses which of the to masses between mercury (mer), venus (ven) and earth (ear) start coming closer (the third point in my question above where the heavier object is noted as making contact first - the earth is the reference ).

From the below output we can see that after starting off with the same distance between them, venus and earth are closer to each other than earth and mercury.

The log file abbreviations are as follows :

the cog-dist - is the distance between the centre of gravity between two masses.

gvt-a-to-b is the acceleration due to gravity - exerted by body a on body b. I mentioned the calculation logic only in the end calculation since the max 30000 chars was reached otherwise.

The mass and diameter references are from here

Start !
GravitationalConstant : 6.67384E-11
Mercury - SurfaceGravity, Mass(x 10^24), Diameter(km) : 3.700735835503674,0.33,4879.0
Venus - SurfaceGravity, Mass(x 10^24), Diameter(km) : 8.873744358760755,4.87,12104.0
Earth - SurfaceGravity, Mass(x 10^24), Diameter(km) : 9.79447341770733,5.97,12756.0
*******************
cogDistVenAndEar : 7500.0
cogDistEarAndMer : 7500.0
cogDistVenAndMer : 15000.0
gvtyVenToEar : 5.778062364444444
gvtyAcclnVenToMer : 1.444515591111111
gvtyAcclnEarToVen : 7.083168853333334
gvtyEarToMer : 7.083168853333334
gvtyMerToEar : 0.39153194666666674
gvtyMerToVen : 0.09788298666666669
*******************
cogDistVenAndEar : 7487.432417742223
cogDistEarAndMer : 7496.858845973334
cogDistVenAndMer : 14984.291263715557
gvtyVenToEar : 5.797475485753492
gvtyAcclnVenToMer : 1.4475458857536994
gvtyAcclnEarToVen : 7.106966868572555
gvtyEarToMer : 7.089105735969546
gvtyMerToEar : 0.39186011605861815
gvtyMerToVen : 0.09808832490733488
*******************
cogDistVenAndEar : 7474.821747179048
cogDistEarAndMer : 7493.727809721305
cogDistVenAndMer : 14968.549556900352
gvtyVenToEar : 5.817053671672689
gvtyAcclnVenToMer : 1.4505921160867346
gvtyAcclnEarToVen : 7.1309672320094375
gvtyEarToMer : 7.095030926914745
gvtyMerToEar : 0.39218763917619204
gvtyMerToVen : 0.09829474297918327
*******************
cogDistVenAndEar : 7462.167619171562
cogDistEarAndMer : 7490.6070527107995
cogDistVenAndMer : 14952.774671882362
gvtyVenToEar : 5.836799182141784
gvtyAcclnVenToMer : 1.453654423231778
gvtyAcclnEarToVen : 7.155172714042394
gvtyEarToMer : 7.100944060456927
gvtyMerToEar : 0.39251449580415176
gvtyMerToVen : 0.09850225044486385
*******************
cogDistVenAndEar : 7449.469659520737
cogDistEarAndMer : 7487.496738913448
cogDistVenAndMer : 14936.966398434186
gvtyVenToEar : 5.856714319853269
gvtyAcclnVenToMer : 1.4567329501725919
gvtyAcclnEarToVen : 7.179586137479264
gvtyEarToMer : 7.106844764621046
gvtyMerToEar : 0.39284066538106294
gvtyMerToVen : 0.09871085699321466
*******************
cogDistVenAndEar : 7436.727488871792
cogDistEarAndMer : 7484.397034853127
cogDistVenAndMer : 14921.124523724919
gvtyVenToEar : 5.876801431286529
gvtyAcclnVenToMer : 1.4598278417892054
gvtyAcclnEarToVen : 7.204210378805046
gvtyEarToMer : 7.112732661056752
gvtyMerToEar : 0.3931661269930868
gvtyMerToVen : 0.09892057244156835
*******************
cogDistVenAndEar : 7423.940722616252
cogDistEarAndMer : 7481.308109654574
cogDistVenAndMer : 14905.248832270827
gvtyVenToEar : 5.897062907772527
gvtyAcclnVenToMer : 1.4629392448928003
gvtyAcclnEarToVen : 7.229048369487063
gvtyEarToMer : 7.11860736492379
gvtyMerToEar : 0.3934908593676467
gvtyMerToVen : 0.09913140673811585
*******************
cogDistVenAndEar : 7411.108970791622
cogDistEarAndMer : 7478.230135093162
cogDistVenAndMer : 14889.339105884785
gvtyVenToEar : 5.917501186590083
gvtyAcclnVenToMer : 1.4660673082614317
gvtyAcclnEarToVen : 7.25410309731885
gvtyEarToMer : 7.124468484775012
gvtyMerToEar : 0.3938148408669605
gvtyMerToVen : 0.09934336996432701
*******************
cogDistVenAndEar : 7398.231837978616
cogDistEarAndMer : 7475.163285645848
cogDistVenAndMer : 14873.395123624465
gvtyVenToEar : 5.938118752094803
gvtyAcclnVenToMer : 1.4692121826766078
gvtyAcclnEarToVen : 7.279377607804102
gvtyEarToMer : 7.130315622436921
gvtyMerToEar : 0.3941380494814379
gvtyMerToVen : 0.0995564723374293
*******************
cogDistVenAndEar : 7385.308923195861
cogDistEarAndMer : 7472.107738543348
cogDistVenAndMer : 14857.41666173921
gvtyVenToEar : 5.958918136881913
gvtyAcclnVenToMer : 1.4723740209607625
gvtyAcclnEarToVen : 7.30487500558214
gvtyEarToMer : 7.13614837288771
gvtyMerToEar : 0.39446046282293884
gvtyMerToVen : 0.09977072421294697
*******************
cogDistVenAndEar : 7372.339819792007
cogDistEarAndMer : 7469.063673823558
cogDistVenAndMer : 14841.403493615566
gvtyVenToEar : 5.979901922984088
gvtyAcclnVenToMer : 1.4755529780156385
gvtyAcclnEarToVen : 7.330598455896305
gvtyEarToMer : 7.141966324132716
gvtyMerToEar : 0.39478205811788886
gvtyMerToVen : 0.09998613608730202
*******************
cogDistVenAndEar : 7359.324115335157
cogDistEarAndMer : 7466.031274386276
cogDistVenAndMer : 14825.355389721433
gvtyVenToEar : 6.001072743105606
gvtyAcclnVenToMer : 1.4787492108616138
gvtyAcclnEarToVen : 7.356551186106873
gvtyEarToMer : 7.147769057077207
gvtyMerToEar : 0.3951028122002476
gvtyMerToVen : 0.10020271860047898
*******************
cogDistVenAndEar : 7346.2613914995445
cogDistEarAndMer : 7463.010726049242
cogDistVenAndMer : 14809.272117548786
gvtyVenToEar : 6.022433281894081
gvtyAcclnVenToMer : 1.4819628786779995
gvtyAcclnEarToVen : 7.382736487250034
gvtyEarToMer : 7.153556145396484
gvtyMerToEar : 0.39542270150432823
gvtyMerToVen : 0.10042048253875563
*******************
cogDistVenAndEar : 7333.151223949367
cogDistEarAndMer : 7460.002217605557
cogDistVenAndMer : 14793.153441554925
gvtyVenToEar : 6.043986277251146
gvtyAcclnVenToMer : 1.485194142844337
gvtyAcclnEarToVen : 7.409157715644628
gvtyEarToMer : 7.159327155403163
gvtyMerToEar : 0.39574170205746123
gvtyMerToVen : 0.1006394388375013
*******************
cogDistVenAndEar : 7319.99318221969
cogDistEarAndMer : 7457.005940882504
cogDistVenAndMer : 14776.999123102194
gvtyVenToEar : 6.06573452168351
gvtyAcclnVenToMer : 1.4884431669827292
gvtyAcclnEarToVen : 7.435818294548369
gvtyEarToMer : 7.165081645911611
gvtyMerToEar : 0.3960597894725011
gvtyMerToVen : 0.10085959858404532
*******************
cogDistVenAndEar : 7306.786829594347
cogDistEarAndMer : 7454.02209080182
cogDistVenAndMer : 14760.808920396168
gvtyVenToEar : 6.087680863695811
gvtyAcclnVenToMer : 1.4917101170012268
gvtyAcclnEarToVen : 7.462721715865297
gvtyEarToMer : 7.170819168099449
gvtyMerToEar : 0.39637693894017056
gvtyMerToVen : 0.10108097302061701
*******************
cogDistVenAndEar : 7293.531722980706
cogDistEarAndMer : 7451.050865441475
cogDistVenAndMer : 14744.582588422181
gvtyVenToEar : 6.109828209226865
gvtyAcclnVenToMer : 1.4949951611383214
gvtyAcclnEarToVen : 7.489871541906445
gvtyEarToMer : 7.176539265366013
gvtyMerToEar : 0.3966931252212369
gvtyMerToVen : 0.10130357354736062
*******************
cogDistVenAndEar : 7280.227412781247
cogDistEarAndMer : 7448.092466098977
cogDistVenAndMer : 14728.319878880224
gvtyVenToEar : 6.132179523130849
gvtyAcclnVenToMer : 1.4982984700085615
gvtyAcclnEarToVen : 7.5172714072055795
gvtyEarToMer : 7.182241473187759
gvtyMerToEar : 0.3970083226385194
gvtyMerToVen : 0.10152741172542615
*******************
cogDistVenAndEar : 7266.873442761824
cogDistEarAndMer : 7445.147097356273
cogDistVenAndMer : 14712.020540118097
gvtyVenToEar : 6.154737830705103
gvtyAcclnVenToMer : 1.5016202166493349
gvtyAcclnEarToVen : 7.544925020392088
gvtyEarToMer : 7.187925318970448
gvtyMerToEar : 0.3973225050687183
gvtyMerToVen : 0.10175249928013974
*******************
cogDistVenAndEar : 7253.469349916515
cogDistEarAndMer : 7442.214967146289
cogDistVenAndMer : 14695.684317062805
gvtyVenToEar : 6.177506219266299
gvtyAcclnVenToMer : 1.5049605765688576
gvtyAcclnEarToVen : 7.572836166123164
gvtyEarToMer : 7.193590321898097
gvtyMerToEar : 0.3976356459340657
gvtyMerToVen : 0.10197884810425525
*******************
cogDistVenAndEar : 7240.014664328955
cogDistEarAndMer : 7439.296286821154
cogDistVenAndMer : 14679.31095115011
gvtyVenToEar : 6.200487839776767
gvtyAcclnVenToMer : 1.5083197277953997
gvtyAcclnEarToVen : 7.601008707077475
gvtyEarToMer : 7.1992359927785525
gvtyMerToEar : 0.3979477181937893
gvtyMerToVen : 0.10220647026128994
*******************
cogDistVenAndEar : 7226.5089090300335
cogDistEarAndMer : 7436.391271222163
cogDistVenAndMer : 14662.900180252196
gvtyVenToEar : 6.2236859085228735
gvtyAcclnVenToMer : 1.5116978509277932
gvtyAcclnEarToVen : 7.629446586012641
gvtyEarToMer : 7.204861833885632
gvtyMerToEar : 0.39825869433538674
gvtyMerToVen : 0.10243537798894699
*******************
cogDistVenAndEar : 7212.9515998518455
cogDistEarAndMer : 7433.500138751538
cogDistVenAndMer : 14646.451738603384
gvtyVenToEar : 6.247103708847428
gvtyAcclnVenToMer : 1.5150951291872614
gvtyAcclnEarToVen : 7.658153827888943
gvtyEarToMer : 7.21046733879772
gvtyMerToEar : 0.39856854636570316
gvtyMerToVen : 0.1026655837026276
*******************
cogDistVenAndEar : 7199.342245277772
cogDistEarAndMer : 7430.623111446034
cogDistVenAndMer : 14629.965356723806
gvtyVenToEar : 6.270744592938168
gvtyAcclnVenToMer : 1.5185117484706165
gvtyAcclnEarToVen : 7.68713454206178
gvtyEarToMer : 7.216051992232729
gvtyMerToEar : 0.39887724580180917
gvtyMerToVen : 0.10289709999903564
*******************
cogDistVenAndEar : 7185.6803462885755
cogDistEarAndMer : 7427.760415052468
cogDistVenAndMer : 14613.440761341044
gvtyVenToEar : 6.29461198367447
gvtyAcclnVenToMer : 1.5219478974048568
gvtyAcclnEarToVen : 7.7163929245455005
gvtyEarToMer : 7.2216152698793
gvtyMerToEar : 0.3991847636616699
gvtyMerToVen : 0.10312993965987739
*******************
cogDistVenAndEar : 7171.965396204358
cogDistEarAndMer : 7424.912279105196
cogDistVenAndMer : 14596.877675309554
gvtyVenToEar : 6.31870937653453
gvtyAcclnVenToMer : 1.525403767403232
gvtyAcclnEarToVen : 7.745933260351365
gvtyEarToMer : 7.227156638224185
gvtyMerToEar : 0.39949107045460325
gvtyMerToVen : 0.10336411565566049
*******************
cogDistVenAndEar : 7158.196880522271
cogDistEarAndMer : 7422.078937005648
cogDistVenAndMer : 14580.27581752792
gvtyVenToEar : 6.343040341565359
gvtyAcclnVenToMer : 1.5288795527227979
gvtyAcclnEarToVen : 7.775759925902505
gvtyEarToMer : 7.232675554375651
gvtyMerToEar : 0.39979613617151843
gvtyMerToVen : 0.10359964114959412
*******************
cogDistVenAndEar : 7144.374276749826
cogDistEarAndMer : 7419.260626103943
cogDistVenAndMer : 14563.63490285377
gvtyVenToEar : 6.367608525418041
gvtyAcclnVenToMer : 1.5323754505235363
gvtyAcclnEarToVen : 7.805877391528894
gvtyEarToMer : 7.238171465882833
gvtyMerToEar : 0.40009993027493057
gvtyMerToVen : 0.10383652950159489
*******************
cogDistVenAndEar : 7130.497054233652
cogDistEarAndMer : 7416.45758778268
cogDistVenAndMer : 14546.954642016331
gvtyVenToEar : 6.392417653450795
gvtyAcclnVenToMer : 1.5358916609290718
gvtyAcclnEarToVen : 7.836290224045431
gvtyEarToMer : 7.24364381055088
gvtyMerToEar : 0.4004024216887422
gvtyMerToVen : 0.10407479427240118
*******************
cogDistVenAndEar : 7116.564673983572
cogDistEarAndMer : 7413.670067542962
cogDistVenAndMer : 14530.234741526534
gvtyVenToEar : 6.417471531902533
gvtyAcclnVenToMer : 1.5394283870890475
gvtyAcclnEarToVen : 7.867003089416452
gvtyEarToMer : 7.249092016251833
gvtyMerToEar : 0.4007035787877898
gvtyMerToVen : 0.10431444922779995
*******************
cogDistVenAndEar : 7102.576588491814
cogDistEarAndMer : 7410.898315092735
cogDistVenAndMer : 14513.474903584549
gvtyVenToEar : 6.442774050139707
gvtyAcclnVenToMer : 1.5429858352432213
gvtyAcclnEarToVen : 7.898020755510071
gvtyEarToMer : 7.25451550073103
gvtyMerToEar : 0.40100336938714237
gvtyMerToVen : 0.10455550834296985
*******************
cogDistVenAndEar : 7088.532241547208
cogDistEarAndMer : 7408.142584437513
cogDistVenAndMer : 14496.67482598472
gvtyVenToEar : 6.468329182979363
gvtyAcclnVenToMer : 1.5465642147873266
gvtyAcclnEarToVen : 7.929348094945955
gvtyEarToMer : 7.2599136714089765
gvtyMerToEar : 0.4013017607311495
gvtyMerToVen : 0.10479798580694412
*******************
cogDistVenAndEar : 7074.431068044208
cogDistEarAndMer : 7405.403133973565
cogDistVenAndMer : 14479.834202017773
gvtyVenToEar : 6.494140993091449
gvtyAcclnVenToMer : 1.550163738340772
gvtyAcclnEarToVen : 7.960990088040237
gvtyEarToMer : 7.265285925178518
gvtyMerToEar : 0.4015987194822297
gvtyMerToVen : 0.10504189602719811
*******************
cogDistVenAndEar : 7060.2724937865305
cogDistEarAndMer : 7402.680226583655
cogDistVenAndMer : 14462.952720370185
gvtyVenToEar : 6.520213633483608
gvtyAcclnVenToMer : 1.5537846218162366
gvtyAcclnEarToVen : 7.992951825851569
gvtyEarToMer : 7.27063164819718
gvtyMerToEar : 0.40189421170939194
gvtyMerToVen : 0.10528725363436513
*******************
cogDistVenAndEar : 7046.05593528527
cogDistEarAndMer : 7399.974129735417
cogDistVenAndMer : 14446.030065020686
gvtyVenToEar : 6.54655135007176
gvtyAcclnVenToMer : 1.5574270844912228
gvtyAcclnEarToVen : 8.025238513332324
gvtyEarToMer : 7.275950215674518
gvtyMerToEar : 0.40218820287648094
gvtyMerToVen : 0.10553407348708493
*******************
cogDistVenAndEar : 7031.780799551255
cogDistEarAndMer : 7397.285115582447
cogDistVenAndMer : 14429.065915133702
gvtyVenToEar : 6.573158484340036
gvtyAcclnVenToMer : 1.5610913490816467
gvtyAcclnEarToVen : 8.057855472589326
gvtyEarToMer : 7.281240991654358
gvtyMerToEar : 0.4024806578301405
gvtyMerToVen : 0.10578237067699045
*******************
cogDistVenAndEar : 7017.446483881478
cogDistEarAndMer : 7394.613461068221
cogDistVenAndMer : 14412.059944949699
gvtyVenToEar : 6.600039476093681
gvtyAcclnVenToMer : 1.5647776418175239
gvtyAcclnEarToVen : 8.090808146258578
gvtyEarToMer : 7.286503328791743
gvtyMerToEar : 0.4027715407874833
gvtyMerToVen : 0.10603216053383635
*******************
cogDistVenAndEar : 7003.0523756393795
cogDistEarAndMer : 7391.959448032919
cogDistVenAndMer : 14395.0118236723
gvtyVenToEar : 6.627198866308814
gvtyAcclnVenToMer : 1.5684861925208298
gvtyAcclnEarToVen : 8.124102100998691
gvtyEarToMer : 7.2917365681244215
gvtyMerToEar : 0.4030608153234605
gvtyMerToVen : 0.10628345863077493
*******************
cogDistVenAndEar : 6988.5978520287645
cogDistEarAndMer : 7389.3233633232585
cogDistVenAndMer : 14377.921215352024
gvtyVenToEar : 6.654641300083057
gvtyAcclnVenToMer : 1.5722172346856216
gvtyAcclnEarToVen : 8.157743031107978
gvtyEarToMer : 7.296940038838767
gvtyMerToEar : 0.40334844435792183
gvtyMerToVen : 0.10653628078978546
*******************
cogDistVenAndEar : 6974.082279861142
cogDistEarAndMer : 7386.7054989054595
cogDistVenAndMer : 14360.7877787666
gvtyVenToEar : 6.682371529691254
gvtyAcclnVenToMer : 1.5759710055604805
gvtyAcclnEarToVen : 8.191736762270388
gvtyEarToMer : 7.302113058029857
gvtyMerToEar : 0.4036343901423539
gvtyMerToVen : 0.10679064308726051
*******************
cogDistVenAndEar : 6959.505015316236
cogDistEarAndMer : 7384.1061519814175
cogDistVenAndMer : 14343.611167297653
gvtyVenToEar : 6.710394417750733
gvtyAcclnVenToMer : 1.5797477462333795
gvtyAcclnEarToVen : 8.226089255435705
gvtyEarToMer : 7.307254930455638
gvtyMerToEar : 0.40391861424629155
gvtyMerToVen : 0.10704656185975674
*******************
cogDistVenAndEar : 6944.865403695436
cogDistEarAndMer : 7381.525625108233
cogDistVenAndMer : 14326.391028803668
gvtyVenToEar : 6.73871494050073
gvtyAcclnVenToMer : 1.5835477017190558
gvtyAcclnEarToVen : 8.260806610839705
gvtyEarToMer : 7.312364948284891
gvtyMerToEar : 0.40420107754338597
gvtyMerToVen : 0.1073040537099155
*******************
cogDistVenAndEar : 6930.162779167928
cogDistEarAndMer : 7378.964226321186
cogDistVenAndMer : 14309.127005489114
gvtyVenToEar : 6.76733819120087
gvtyAcclnVenToMer : 1.5873711210489698
gvtyAcclnEarToVen : 8.295895072170268
gvtyEarToMer : 7.317442390838887
gvtyMerToEar : 0.4044817401971245
gvtyMerToVen : 0.10756313551255856
*******************
cogDistVenAndEar : 6915.396464509242
cogDistEarAndMer : 7376.4222692603025
cogDistVenAndMer : 14291.818733769545
gvtyVenToEar : 6.796269383653805
gvtyAcclnVenToMer : 1.5912182573639673
gvtyAcclnEarToVen : 8.331361030885672
gvtyEarToMer : 7.322486524326471
gvtyMerToEar : 0.40476056164618684
gvtyMerToVen : 0.107823824420967
*******************
cogDistVenAndEar : 6900.565770831928
cogDistEarAndMer : 7373.90007330062
cogDistVenAndMer : 14274.465844132548
gvtyVenToEar : 6.825513855857396
gvtyAcclnVenToMer : 1.5950893680097205
gvtyAcclnEarToVen : 8.367211030691717
gvtyEarToMer : 7.3274966015723875
gvtyMerToEar : 0.4050375005894285
gvtyMerToVen : 0.10808613787334863
*******************
cogDistVenAndEar : 6885.669997308094
cogDistEarAndMer : 7371.397963686305
cogDistVenAndMer : 14257.067960994398
gvtyVenToEar : 6.855077073792056
gvtyAcclnVenToMer : 1.5989847146350686
gvtyAcclnEarToVen : 8.403451772184512
gvtyEarToMer : 7.332471861738631
gvtyMerToEar : 0.40531251497047716
gvtyMerToVen : 0.10835009359950158
*******************
cogDistVenAndEar : 6870.708430883488
cogDistEarAndMer : 7368.916271668753
cogDistVenAndMer : 14239.624702552243
gvtyVenToEar : 6.884964635349187
gvtyAcclnVenToMer : 1.6029045632933552
gvtyAcclnEarToVen : 8.440090117666252
gvtyEarToMer : 7.3374115300385805
gvtyMerToEar : 0.40558556196193163
gvtyMerToVen : 0.10861570962768116
*******************
cogDistVenAndEar : 6855.680345982807
cogDistEarAndMer : 7366.455334648809
cogDistVenAndMer : 14222.135680631616
gvtyVenToEar : 6.915182274406945
gvtyAcclnVenToMer : 1.6068491845468889
gvtyAcclnEarToVen : 8.477133096141575
gvtyEarToMer : 7.342314817443706
gvtyMerToEar : 0.40585659794914963
gvtyMerToVen : 0.10888300429167833
*******************
cogDistVenAndEar : 6840.585004205916
cogDistEarAndMer : 7364.015496323276
cogDistVenAndMer : 14204.600500529192
gvtyVenToEar : 6.945735865059849
gvtyAcclnVenToMer : 1.6108188535746242
gvtyAcclnEarToVen : 8.514587908502527
gvtyEarToMer : 7.347180920382562
gvtyMerToEar : 0.406125578513609
gvtyMerToVen : 0.10915199623811625
*******************
cogDistVenAndEar : 6825.421654014629
cogDistEarAndMer : 7361.597106835865
cogDistVenAndMer : 14187.018760850493
gvtyVenToEar : 6.9766314260091224
gvtyAcclnVenToMer : 1.6148138502832086
gvtyAcclnEarToVen : 8.552461932910568
gvtyEarToMer : 7.352009020431848
gvtyMerToEar : 0.4063924584158308
gvtyMerToVen : 0.10942270443397513
*******************
cogDistVenAndEar : 6810.1895304096915
cogDistEarAndMer : 7359.200522932742
cogDistVenAndMer : 14169.390053342435
gvtyVenToEar : 7.007875125120996
gvtyAcclnVenToMer : 1.6188344594215067
gvtyAcclnEarToVen : 8.590762730384466
gvtyEarToMer : 7.356798283999241
gvtyMerToEar : 0.4066571915778476
gvtyMerToVen : 0.10969514817435262
*******************
cogDistVenAndEar : 6794.88785459759
cogDistEarAndMer : 7356.826108122866
cogDistVenAndMer : 14151.713962720456
gvtyVenToEar : 7.039473284160591
gvtyAcclnVenToMer : 1.6228809706987513
gvtyAcclnEarToVen : 8.629498050603436
gvtyEarToMer : 7.361547861997748
gvtyMerToEar : 0.4069197310652022
gvtyMerToVen : 0.10996934709046981
*******************
cogDistVenAndEar : 6779.515833646801
cogDistEarAndMer : 7354.474232843264
cogDistVenAndMer : 14133.990066490065
gvtyVenToEar : 7.071432383709382
gvtyAcclnVenToMer : 1.626953678906447
gvtyAcclnEarToVen : 8.668675837935321
gvtyEarToMer : 7.366256889511263
gvtyMerToEar : 0.4071800290684618
gvtyMerToVen : 0.11024532115793173
*******************
cogDistVenAndEar : 6764.072660133067
cogDistEarAndMer : 7352.145274629486
cogDistVenAndMer : 14116.217934762553
gvtyVenToEar : 7.103759068274641
gvtyAcclnVenToMer : 1.6310528840441865
gvtyAcclnEarToVen : 8.708304237700126
gvtyEarToMer : 7.370924485451032
gvtyMerToEar : 0.40743803688422797
gvtyMerToVen : 0.11052309070525292
*******************
cogDistVenAndEar : 6748.5575117732715
cogDistEarAndMer : 7349.839618291381
cogDistVenAndMer : 14098.397130064652
gvtyVenToEar : 7.136460151599796
gvtyAcclnVenToMer : 1.6351788914495287
gvtyAcclnEarToVen : 8.748391602679833
gvtyEarToMer : 7.375549752202722
gvtyMerToEar : 0.4076937048956279
gvtyMerToVen : 0.11080267642265802
*******************
cogDistVenAndEar : 6732.969551047466
cogDistEarAndMer : 7347.557656094433
cogDistVenAndMer : 14080.527207141899
gvtyVenToEar : 7.169542622185015
gvtyAcclnVenToMer : 1.639332011932096
gvtyAcclnEarToVen : 8.788946499885943
gvtyEarToMer : 7.3801317752637505
gvtyMerToEar : 0.40794698255226763
gvtyMerToVen : 0.11108409937116874
*******************
cogDistVenAndEar : 6717.307924808575
cogDistEarAndMer : 7345.299787946869
cogDistVenAndMer : 14062.607712755445
gvtyVenToEar : 7.203013649027896
gvtyAcclnVenToMer : 1.6435125619120725
gvtyAcclnEarToVen : 8.829977717596828
gvtyEarToMer : 7.384669622870545
gvtyMerToEar : 0.4081978183496281
gvtyMerToVen : 0.11136738099198851
*******************
cogDistVenAndEar : 6701.571763879308
cogDistEarAndMer : 7343.0664215927645
cogDistVenAndMer : 14044.638185472071
gvtyVenToEar : 7.236880587594612
gvtyAcclnVenToMer : 1.6477208635632623
gvtyAcclnEarToVen : 8.871494272677584
gvtyEarToMer : 7.389162345615346
gvtyMerToEar : 0.4084461598078835
gvtyMerToVen : 0.11165254311619645
*******************
cogDistVenAndEar : 6685.760182635727
cogDistEarAndMer : 7340.857972811373
cogDistVenAndMer : 14026.6181554471
gvtyVenToEar : 7.271150986032545
gvtyAcclnVenToMer : 1.6519572449609097
gvtyAcclnEarToVen : 8.913505418195953
gvtyEarToMer : 7.39360897605222
gvtyMerToEar : 0.40869195345012277
gvtyMerToVen : 0.1119396079747639
*******************
cogDistVenAndEar : 6669.872278576973
cogDistEarAndMer : 7338.674865622943
cogDistVenAndMer : 14008.547144199916
gvtyVenToEar : 7.305832591635904
gvtyAcclnVenToMer : 1.6562220402344696
gvtyAcclnEarToVen : 8.956020651348327
gvtyEarToMer : 7.398008528291852
gvtyMerToEar : 0.4089351447799517
gvtyMerToVen : 0.11222859820890659
*******************
cogDistVenAndEar : 6653.90713188056
cogDistEarAndMer : 7336.517532501272
cogDistVenAndMer : 13990.424664381833
gvtyVenToEar : 7.340933357576565
gvtyAcclnVenToMer : 1.6605155897255275
gvtyAcclnEarToVen : 8.999049721710904
gvtyEarToMer : 7.4023599975847345
gvtyMerToEar : 0.40917567825845275
gvtyMerToVen : 0.11251953688078525
*******************
cogDistVenAndEar : 6637.86380494265
cogDistEarAndMer : 7334.386414593279
cogDistVenAndMer : 13972.25021953593
gvtyVenToEar : 7.376461449913014
gvtyAcclnVenToMer : 1.6648382401510922
gvtyAcclnEarToVen : 9.042602639831765
gvtyEarToMer : 7.406662359892301
gvtyMerToEar : 0.409413497280479
gvtyMerToVen : 0.11281244748457095
*******************
cogDistVenAndEar : 6621.741341902701
cogDistEarAndMer : 7332.281961945869
cogDistVenAndMer : 13954.02330384857
gvtyVenToEar : 7.412425254891002
gvtyAcclnVenToMer : 1.6691903447724772
gvtyAcclnEarToVen : 9.086689686180552
gvtyEarToMer : 7.410914571445574
gvtyMerToEar : 0.40964854415025787
gvtyMerToVen : 0.11310735395788862
*******************
cogDistVenAndEar : 6605.538768151821
cogDistEarAndMer : 7330.204633740391
cogDistVenAndMer : 13935.743401892212
gvtyVenToEar : 7.448833386550293
gvtyAcclnVenToMer : 1.673572263570014
gvtyAcclnEarToVen : 9.131321420473357
gvtyEarToMer : 7.415115568290872
gvtyMerToEar : 0.4098807600562793
gvtyMerToVen : 0.11340428069365603
*******************
cogDistVenAndEar : 6589.255089824161
cogDistEarAndMer : 7328.154898535024
cogDistVenAndMer : 13917.409988359184
gvtyVenToEar : 7.4856946946527225
gvtyAcclnVenToMer : 1.6779843634238385
gvtyAcclnEarToVen : 9.17650869139153
gvtyEarToMer : 7.419264265822032
gvtyMerToEar : 0.410110085045439
gvtyMerToVen : 0.11370325255233406
*******************
cogDistVenAndEar : 6572.88929327061
cogDistEarAndMer : 7326.133234515385
cogDistVenAndMer : 13899.022527785995
gvtyVenToEar : 7.523018272947625
gvtyAcclnVenToMer : 1.6824270183010135
gvtyAcclnEarToVen : 9.22226264671403
gvtyEarToMer : 7.423359558298708
gvtyMerToEar : 0.410336457996411
gvtyMerToVen : 0.11400429487460667
*******************
cogDistVenAndEar : 6556.44034451407
cogDistEarAndMer : 7324.140129753737
cogDistVenAndMer : 13880.580474267806
gvtyVenToEar : 7.560813467791674
gvtyAcclnVenToMer : 1.6869006094492605
gvtyAcclnEarToVen : 9.26859474388425
gvtyEarToMer : 7.427400318350135
gvtyMerToEar : 0.4105598165922186
gvtyMerToVen : 0.11430743349450842
*******************
cogDistVenAndEar : 6539.907188685492
cogDistEarAndMer : 7322.176082477136
cogDistVenAndMer : 13862.083271162628
gvtyVenToEar : 7.59908988714107
gvtyAcclnVenToMer : 1.6914055255975853
gvtyAcclnEarToVen : 9.315516761033306
gvtyEarToMer : 7.431385396463854
gvtyMerToEar : 0.4107800972919719
gvtyMerToVen : 0.11461269475301915
*******************
cogDistVenAndEar : 6523.288749439856
cogDistEarAndMer : 7320.241601344925
cogDistVenAndMer : 13843.53035078478
gvtyVenToEar : 7.637857409935262
gvtyAcclnVenToMer : 1.6959421631641043
gvtyAcclnEarToVen : 9.36304080848327
gvtyEarToMer : 7.435313620458762
gvtyMerToEar : 0.41099723530174065
gvtyMerToVen : 0.11492010551214671
*******************
cogDistVenAndEar : 6506.583928351227
cogDistEarAndMer : 7318.337205735935
cogDistVenAndMer : 13824.921134087162
gvtyVenToEar : 7.677126195892252
gvtyAcclnVenToMer : 1.7005109264713805
gvtyAcclnEarToVen : 9.41117934075498
gvtyEarToMer : 7.439183794941917
gvtyMerToEar : 0.4112111645445281
gvtyMerToVen : 0.11522969316951863
COG - Venus, Earth : 6506.583928351227 - 7.677126195892252 - 9.41117934075498 - 0.11522969316951863 + 0.4112111645445281
COG - Earth, Mercury : 7318.337205735935 - 7.439183794941917 - 0.4112111645445281 - 1.7005109264713805 + 7.677126195892252
Distance limit reached :6500.0
Done !

• The output here is completely meaningless without the code itself (for us to run & verify and/or criticize). If it is too long, consider posting the code at a code hosting site like github – Kyle Kanos Mar 29 '15 at 17:33
• I dunno, this isn't a peer-reviewed venue, I don't think we necessarily need to see source code for all results posted. That being said, it certainly wouldn't hurt. The bigger issue I think is that long lists of numbers are not useful. This would be better off as a visualization of some kind. – David Z Mar 29 '15 at 18:21
• @KyleKanos will try posting on github. But all the details are there in the post above for whoever is interested in verifying the same by writing their own code. – Ravindra HV Apr 2 '15 at 19:19
• @DavidZ it'd be easier for me to post the code than turn it into a graph. Will post it if I make one ! – Ravindra HV Apr 2 '15 at 19:20
• @KyleKanos Here is the github link for the java source(github.com/ravindrahv/SOSamples/blob/master/src/com/example/so/…) – Ravindra HV Apr 2 '15 at 20:07

## protected by Qmechanic♦Oct 4 '15 at 7:36

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