Of course, the next thing needed is a rule to understand how an object changes its speed when something affects it. That is, Newton`s contribution. Newton wrote three laws: The first law was a simple reformulation of the Galilean principle of inertia just described. The second law gave a specific way to determine how velocity changes under various influences called forces. The third act describes the forces to some extent, and we will discuss that at another time. Here we will only discuss the second law of the law, which states that the motion of an object is modified by forces in this way: the rate of temporal change of a quantity called quantity called momentum is proportional to the force. We`ll figure it out mathematically shortly, but let`s explain the idea first. I struggled with this information again, mainly because I thought Lisa had already ended all contact with Tyler. Our family had lived the last few months assuming that they were safe from its influence, that they were ultimately exaggerated. She had sworn it was over. She promised it. Our calculation therefore proceeds in the following steps using the time intervals $epsilon=0.100$: Initial values at $t=0$: begin{alignat*}{2} x(0)&=0.500&qquadqquad y(0)&=phantom{+}0.000[.5ex] v_x(0)&=0.000&qquadqquad v_y(0)&=+1.630 end{alignat*} Among these we find: begin{alignat*}{2} r(0)&=phantom{-}0.500&qquad 1/r^3(0)&=8.000[.5ex] a_x(0)&=-4.000&qquad a_y(0)&=0.000 end{alignat*} How to calculate Speeds $v_x(0.05)$ and $v_y(0.05)$: begin{align*} v_x(0.05) &= 0.000 – 4.000 times 0.050 = -0.200;[1ex] v_y(0.05) &= 1.630 + 0.000 times 0.050 = phantom{-}1.630. end{align*} Now our main calculations begin: begin{alignat*}{2} x(0.1)&=0.500-0.20 times 0.1&&=phantom{-}0.480[.5ex] y(0.1)&=0.0+1.63 times 0.1 &&=phantom{-}0.163[.5ex] r(0.0.1) r(0.1) 1)&=5ex) 1)&=sqrt{0.0.480^2+0.163^2}&&=phantom{-}0.507[.5ex] 1/r^3(0.1)&=7.677 &&[.5ex] a_x(0.1)&=-0.480 times 7.677 &&=-3.685[.5ex] a_y(0.1)&=-0.163 times 7.677 &&=-1.250[.5ex] v_x(0.15)&=-0.200-3.685times0.1 &&=-0.568[.5ex] v_y(0.15)&=1.630-1.250times0.1 &&&= phantom{-}1.505 [.5ex] x( 0.2)&=0.480-0.568times 0.1&&=phantom{-}0.423[.5ex] y(0.2)&=0.163+1.505times0.1&&=phantom{-}0.313[.5ex] &qquadqquadtext{etc.}&& end{alignat*} This gives us the values given in Table 9–2, and in increments of about $20, we chased the planet halfway around the sun! Figures 9 to 6 show the coordinates $x$ and $$y$ shown in Table 9–2.
The dots represent positions in the sequence of times separated by one-tenth of a unit; We see that the planet moves rapidly at the beginning and slowly at the end, and thus the shape of the curve is determined. So we see that we really know how to calculate the movement of the planets! Now we are ready to do our calculation. To simplify, we can organize the work in the form of a table, with columns for time, position, speed and acceleration, and intermediate lines for speed, as shown in table 9-1. Such a table is, of course, only a practical way to represent the numerical values obtained from the set of equations (9.16), and in fact the equations themselves never need to be written. We simply fill in the different fields of the table one by one. This painting now gives us a very good idea of the movement: it starts from rest, first takes a slight upward (negative) speed and loses part of its distance. The acceleration is then slightly weaker, but still gains speed. But over time, it picks up speed faster and slower until we can confidently predict at $x=$0 to about $t=$1.50 sec that it will continue, but now it will be on the other side; The $x$ position becomes negative, so the acceleration becomes positive. As a result, the speed decreases. It is interesting to compare these numbers with the function $x=cos t$ performed in Fig.
9–4. The match is in the three-digit accuracy of our calculation! We will see later that $x=cos t$ is the exact mathematical solution of our equation of motion, but it is a powerful example of the power of numerical analysis that such a simple calculation gives such accurate results. It was only with the advent of modern physics in the early 20th century that it was discovered that Newton`s laws of motion only produce a good approximation of motion if objects move at speeds much, much lower than the speed of light, and if these objects are larger than the size of most molecules (about 10-9 m10-9 m in diameter). These limits define the field of classical mechanics, as discussed in Introduction to the Nature of Science and Physics.
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