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.
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:
Those four possibility are made visible in the simulations of the program 2OBJECTS. In this program we see how two objects behave.
Next perform the programs: 2OBJECTS.TXT 2.1 TEST 1
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.
Next perform the programs:
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:
m1 ai1 :: ------ 3.6 rērē = the distance in quadrate between object i and 1
axi1 is the acceleration in the x direction between object i and 1
rx axi1 = ai1 * -- 3.7 rrx = distance between object i and 1 in the x direction
axi is the sum of axij for j = from 1 to n except for j = i
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
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
saxi is the sum of axi
svxi is the sum of vxin
Equation 3.8 now becomes:
vxi0 is the velocity of object i at t = 0
xin is xi0 + svxi * dt 3.11
xi0 is the position of object i at t = 0
Return back to INDEX.TXT