This chapter discusses Newton's Law.

Topics described are gravity.

To explain those topics the programs DROP, 2OBJECTS and 3OBJECTS are used.

In order to understand Newton's Law we are going to perform the following test:
Take up a stone from the ground and let the stone loss. What you will see is that the stone will fall and after some time hit the ground.

In actual fact Isaac Newton (at least is that a story) more or less did the same thing: he studied how an apple falls from a tree.

Only dropping a stone is not enough. While the stone is falling you must perform the following observations: measure at regular intervals the height of the stone above the ground.

The results you write down in a table.

For example you can get the following (normalised) results:

                   time       height
                     0           16
                     1           15
                     2           12
                     3            7
                     4            0

With those values you are going to calculate two values: the speed of the stone and the acceleration of the stone.

From chapter 1 formula (1.4) we know that speed is equal to:

            s2 - s1        delta  s
        v = -------    =   --------                     (2.1)
            t2 - t1        delta  t

From chapter 1 formula (1.5) we know that acceleration is equal to:

            v2 - v1        delta  v
        a = -------    =   --------                     (3.1)
            t2 - t1        delta  t

With those 2 formulas we are going to calculate the speed and acceleration with the small difference that for position we use height:

   ------------------------------------------------------------------------
                            delta      delta                 delta     
      time     distance     time      distance      v          v         a
   ------------------------------------------------------------------------
        0          0
        1          1          1          1          1
        2          4          1          3          3          2         2
        3          9          1          5          5          2         2
        4         16          1          7          7          2         2
   ------------------------------------------------------------------------

The above table demonstrates that the speed v is increasing and that the acceleration a is constant during the fall.

Now perform the program: DROP.TXT

The fact that during the whole fall there is acceleration and that the acceleration is constant is remarkable. It even more remarkable when you realise that all bodies fall with (have) the same acceleration. (See also equation 3.4 below)

That is also what Isaac Newton thought. He went one step further and he postulated the following law:
Whenever there is acceleration there always is a force, which is the cause.

In our case it is the Earth (mass) that puts a force on the stone i.e. attracts the stone

He even went one step further:
All masses put forces on each other. That is Newton's Law.

For two masses this force is equivalent with the masses of the two bodies subdivided by the square of the distance between the two masses

                m1 * m2

           F :: -------                         3.1

                   rē

This means that m1 is attracted towards m2 with a force F and m2 is attracted towards m2 also with the same force F.

The force F can also be written as:

F = m1 * a1 3.2

a1 = acceleration of m1

or F = m2 * a2 3.3

a2 = acceleration of m2

                   m2
           a1 :: ------                                3.4
                   rē

And for acceleration a2:

                   m1
           a2 :: ------                                3.5
                   rē

In words equation 3.4 and 3.5 tell you:

1. When two masses m1 and m2 are equal that the acceleration of both masses will be equal and also the speed.
2. That when the masses are different and m1 is the greatest that then a2 will greater then a1. This means that the speed of m2 will be the greater then the speed of m1.
3. That when the mass of m2 is almost 0 that the acceleration of m1 is close to zero and also the speed of m1.
4. The smaller the distance is between the two masses the larger the acceleration is. This means that the speed increases the closer the 2 masses approach each other and even more then when acceleration would be constant.

Those four possibility are made visible in the simulations of the program 2OBJECTS. In this program we see how two objects behave.

First read the Introduction: 2OBJECTS.TXT

Next perform the programs: 2OBJECTS.TXT 2.1 TEST 1

until: 2OBJECTS.TXT 2.8 TEST 8

Newton's law gives an accurate description how the stars and planets move in our universe. Most of the planets move in circles around the Sun. Simulations of how planets move is demonstrated in the program 3OBJECTS.

First read the introduction: 3OBJECTS.TXT

Next perform the programs:

3OBJECTS.TXT 2.1 TEST 1,
3OBJECTS.TXT 2.2 TEST 2,
3OBJECTS.TXT 2.4 TEST 4,
3OBJECTS.TXT 2.5 TEST 5,
3OBJECTS.TXT 2.9 TEST 9,
3OBJECTS.TXT 2.13 TEST 13

Those tests simulate the movement of stars and planets in straight lines or in circles.

The only exception is the planet Mercury. The planet Mercury moves in an ellipse around the Sun. That is the topic of the chapter 4.

Return back to INDEX.TXT

The following is an option

In program 2OBJECTS the next position xin of object i at t = n is calculated in the following sequence of steps:

1. calculate ai1.
   ai1 is the acceleration between object i and object 1

                       m1
              ai1 :: ------                         3.6
                       rē
   

   rē = the distance in quadrate between object i and 1

2. calculate axi1
   

   axi1 is the acceleration in the x direction between object i and 1

                           rx
             axi1 = ai1 *  --                       3.7
                            r 
   

   rx = distance between object i and 1 in the x direction

3. calculate axi

   axi is the sum of axij for j = from 1 to n except for j = i

4. calculate vxin

   vxin is the velocity of object i at t = n
   vxin-1 is the velocity of object i at t = n-1
   dt is the time between calculations

      vxin is vxin-1 + axi * dt                          3.8
   

5. 5. calculate xin

   xin is the position of object i at t = n
   xin-1 is the position of object i at t = n-1
   

      xin is xin-1 + vxin * dt                           3.9
   

For yin (and zin) similar calculations are performed

In the programs 3OBJECTS, MERCURY, PLANETS and PLANET3D a different strategy is followed. The main reason is accuracy.

Equation 3.8 becomes less accurate the larger vxin becomes compared to axi
Equation 3.9 becomes less accurate the larger xin becomes compared to vxin

To solve this individual accumulators are used:

saxi is accumulator for axi
svxi is accumulator for vxin etc. for y (and z)

saxi is the sum of axi
svxi is the sum of vxin

Equation 3.8 now becomes:

vxin is vxi0 + saxi * dt 3.10

vxi0 is the velocity of object i at t = 0

Equation 3.9 now becomes:

xin is xi0 + svxi * dt 3.11

xi0 is the position of object i at t = 0

Return back to INDEX.TXT