This results that also v0 of m0 = v1 of m1

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 2

The time it takes before m0 and m1 collide is 786 seconds.

Test 3 is almost identical as test 1 i.e. The mass of m0 is large i.e. much greater then m1 The mass of m1 is small i.e. much smaller then m0 The initial speed of both m0 and m1 are zero.

The only difference is that both m0 and m1 have an identical upwards speed relative to the connecting line between m0 and m1.

The simulation shows that the two masses m0 and m1 will move towards each other as in test 1 but the trajectory followed is different.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 3

The time before collision in both test 1 and test 3 is 1110 seconds. i.e. the speed upwards has no influence.

Newton's law describes this, because the force is dependent on distance and masses which are identical in test 1 and 3. The force is independent on (the direction of) speed.

The same is true for the test 5 and 7.

Test 4 is almost identical as test 2 i.e. In this test both m0 and m1 are equal. The initial speed of both m0 and m1 are zero.

The only difference is that both m0 and m1 have an identical upwards speed relative to the connecting line between m0 and m1.

The simulation shows that the two masses m0 and m1 will move towards each other as in test 1 but the trajectory followed is different.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 4

The time before collision in both test 2 and test 4 is 786 seconds. i.e. the speed upwards has no influence.

Newton's law describes this, because the force is dependent on distance and masses which are identical in test 2 and 4. The force is independent on (the direction of) speed.

The same is true for the test 6 and 8.

Test 5 is almost identical as test 3

The only difference is that both m0 and m1 have an identical speed component to the right, relative to the connecting line between m0 and m1.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 5

The time before collision in each test 1, test 3 and test 5 is 1110 seconds. i.e. the speed to the right has no influence.

Test 6 is identical as test 8 and is not implemented.

Test 7 is identical as test 5, except that the speed component is to the left.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 7

The time before collision in each test 1, test 3, test 5 and test 7 is 1110 seconds. i.e. the speed to the left has no influence.

Test 8 is almost identical as test 4.

The only difference is that both m0 and m1 have an identical speed component to the left, relative to the connecting line between m0 and m1.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 8

The time before collision in each test 2, test 4 and test 8 is 786 seconds. i.e. the speed to the left has no influence.

After performing test 8:

Return back to CHAPTER2.TXT

Test 9 is identical as test 1 except that the concept "that propagation of gravitation takes time" is included.

The display shows both the results of test 1 on the top part and of test 9 on the bottom part. The bottom part shows besides a trail of dots, representing the position at each instant, also a white dot. The white dot represents the virtual position. The concept of virtual position is explained in chapter 3. The virtual position represents the position at an earlier moment. How much earlier depends on the time it takes for the propagation to travel between the two objects. The time difference between "now" and this earlier moment is equal to the distance between the two objects divided by the speed of light.

The results of test 1 and test 9 are identical. In this special case propagation time has no influence. The reason why there is no difference is because m0 is heavy (compared to m1) and as a consequence m0 does not move in test 1 and in test 9.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 9

Because m0 does not move the force that m1 feels in test 9 is based on the true distance, which is the same as in test 1.

The time before collision in test 1 and test 9 is both 1110 seconds

Test 10 is identical as test 2 except that the concept "that propagation of gravitation takes time" is included.

In this case both objects move compared to only one in test 9

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 10

The time between start and collision in test 2 is 786 seconds and in test 10 is 815 seconds

The reason is because the distance based on the virtual position is larger then the true distance (based on the true position). As a result the acceleration (and speed) of each object will be smaller. This in turn will give a longer time before collision.

Compare test 10 with test 12 and test 16.

Test 11 is identical as test 3 except that the concept "that propagation of gravitation takes time" is included.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 11

The time between start of test and collision in test 3 is 1110.2 seconds. In test 11 this is 1110.6 seconds

The results are almost identical. The reason is because the distance between and m1 and the virtual position of m0 and the true distance is almost identical.

Compare test 11 with test 13 and test 15

Test 12 is identical as test 4 except that the concept "that propagation of gravitation takes time" is included.

The time between start of test and collision in test 4 is 786 seconds. In test 12 this is 815 seconds

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 12

Compare test 12 with test 10 and test 16.

The results are identical. This is because the virtual positions are identical in each of those tests and the true positions are identical, resulting the same time periods. The direction of the base speed is different but this has no influence on the outcome of the experiment.

Test 13 is identical as test 5 except that the concept "that propagation of gravitation takes time" is included.

The time between start and collision in test 5 is 1110 seconds and in test 13 is 1127 seconds

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 13

Test 13 is special because it has a speed component to the right. As a result is the virtual position of m0 to the left. This in turn means that the virtual distance is larger as the true distance, resulting in a lower acceleration and speed and in a longer time before to collide.

Compare test 13 with test 11 and test 15.

Test 14 is identical as test 16 and is not included.

Test 15 is identical as test 7 except that the concept "that propagation of gravitation takes time" is included.

The time between start and collision in test 7 is 1110 seconds and in test 10 is 1094 seconds

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 15

Test 15 is special because it has a speed component to the left. As a result is the virtual position of m0 to the right. This in turn means that the virtual distance is smaller as the true distance, resulting in a larger acceleration and speed and in a shorter time before to collide.

Compare test 13 with test 11 and test 15.

Test 16 is identical as test 8 except that the concept "that propagation of gravitation takes time" is included.

The time between start and collision in test 8 is 786 seconds and in test 16 is 815 seconds

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 16

Compare test 16 with test 10 and test 12.

The results are identical. This is because the virtual positions are identical in each of those tests and the true positions are identical, resulting the same time periods. The direction of the base speed is different but this has no influence on the outcome of the experiment.

After performing test 16:

Return back to CHAPTER3.TXT

The purpose of this test is to demonstrate how we should describe the reality: doing predictions, solely based on observations.

A standard 2 body simulation starts with the following ingredients: 2 masses, 4 initial conditions (i.e. the position and velocity of both masses at t0) and Newton's law.
Newton's law services as the model to do our predictions.

In this simulation a slightly different approach is taken.

• First, solely based on observations, the positions of both m0 and m1 are established at three (equally spaced) moments: t0, t1 and t2.
  
  The 3 positions for m0 are called: p0t0, p0t1 and p0t2
  The 3 positions for m1 are called: p1t0, p1t1 and p1t2

  Next v0 of m0 at t0 and v1 of m0 at t1 are calculated using:

            |v0| = |p0t0 - p0t1|       |v1| = |p0t1 - p0t2|
  

  Next acceleration a0 of m0 is calculated using:

            |a0| = |v0| - |v01| = |p0t0 -2*p0t1 + p0t2|
  

  Next m1 is calculated using:

   
           G*m0*m1
           -------  = |a0| * m0
              rē
  
                 m1 = |a0|* rē / G
  

  The same is done for acceleration a1 of m1 and m0:

            |a1| = |p1t0 -2*p1t1 + p1t2|
  
              m0 = |a1|* rē / G
  

  Next the speed v2 at t2 for both m0 and m1 are calculated.

  Finally the program calculates the positions of both m0 and m1, starting with t3 using Newton's Law as a model.

In this two body simulation you can only change the position of m1. The position of m0 is modified equivalent.
When you start the simulation the top right side shows the position of m1 at t0, t1 and t2. The position of t0 is in red (being selected). The position of t1 and t2 are in red.

The simulation goes in two steps:

1. First select the position you want to modify
   In order to select the position (one out of three) use the Tab key.
   Second change the position.

   In order to change the position use the arrow keys.

• After the Tab key is pressed the new selected position will turn in red and the old one will change in white.
• After any arrow key is pressed the program first calculates the mass of the two objects m0 and m1 and the predicted positions, starting from t3, are shown.

Now perform the program: 2OBJECTS.EXE

From the Test Selection Display:

Select test 17

It should be mentioned that the calculation in the program is not correct.

• The first reason is because the observations are arbitrary and not real ones.
• The second reason is that the calculation is based at equally spaced moments. This implies that every time at least two positions have to be changed (and not one) before the calculation is performed.
• The third reason is that no other object is included i.e. no other forces influence this simulation.

The main message of the test that the reality is based on observations only. Using those observations the future can be predicted (calculated). In this test this is only done based on Newton's Law i.e. objects have masses concentrated in one point and forces act instanteneous. It is the purpose of this book to show that this model is to simple.

Return back to PROVE.TXT

The program uses the following standard feature: