II Exploring the world of motions: the simplest approach
Having convinced ourselves about the efficiency of numerical iterations in approximating the exact motion, we turn to examples which are not part of the school curriculum and provide a new insight into different aspects of physics. Let us recognize first that with a known form of acceleration $a$, which the rate of change, or trend, of the velocity, iteration (3) can be used also to obtain the velocity at discrete time instants $t=n \Delta t$: $$ v_{n+1} = v_n + a_n\Delta t. \qquad (4) $$ Here $a_n$ denotes the acceleration $a$ of the body taken at time instant $n \Delta t$. In the spirit of Newtons's law, the acceleration is just the total force divided by the mass of the moving body (for simplicity we do not investigate systems consisting of several bodies). If you know the resultant force as a function of position, velocity and time, function $a$ is immediate to derive. This was you are able to convert the Newtonian equation of motion into an iteration of the from of (4) which can always be solved numerically. In our first examples, the acceleration shall be a function of the velocity only. The blue links below lead to Excel worksheets prepared for monitoring the given motions, and to related problems and tasks.
Parachuting: the quadratic air drag leads is proportional to $v^2$, the acceleration caused by it can be written as $kv^2$, with $k$ as a constant. This acceleration should be subtracted from the downward pointing gravity $g$, thus the total acceleration is $$ a = g - kv^2. $$ Monitor this motion with $k=0.2\ 1/m$.
Vertical projection in air. Study the effect of air resistance on vertical projection (the same we started with in module I) numerically by means of an Excel worksheet. The acceleration due to the drag points in the same direction as gravity during rising, and in opposite direction during falling: $$ a=-g-k |v| v $$ (naturally upward direction is considered to be positive here). Use the $k$ value $k=0.1\ 1/\text{m}$ of a light ball.
Free kicks.
We are interested mainly in the path of the soccer ball. To this end we have to follow the motion both in the horizontal
($x$) and in the vertical ($y$) direction. Accordingly, two iterations are needed, for the two velocity components $v_x$
and $v_y$. The corresponding acceleration components are
$$ a_x = -g-k |v| v_x , \quad a_y = -g-k |v| v_y $$
with $v$ as the modulus $(v_x^2+v_y^2)^{1/2}$. Analogous ones hold for the $x$ and $y$ coordinates in the plane of the path.
A good soccer player can kick the ball with an initial velocity $v_0=40 \text{ m/s}$. Assume that the motion starts under an
angle of 15° so that $v_{x,0} = 38.6 \text{ m/s}$, $v_{x,0} = 10.4 \text{ m/s}$, and use the $k$ value (0.022 1/m) determined
for standard soccer balls. Monitor the motion.
General motions. Acceleration, as a rule, also depends on the position, since force $F$ governing the motion typically depends on $x$ too, according to Newton’s law $a(x,v)=F(x,v)/m.$ One should then monitor location and velocity simultaneously. (It is, in general,impossible to unfold velocity alone without being interested in the position, as e.g. in the parashuting problem). We therefore have to turn to a parallel iteration of $v$ and $x$. Since the rate of change of velocity equals acceleration $a$, while that of $x$ is the velocity itself, we obtain the iteration (called the Euler method) $$ v_{n+1} = v_n + a_n\Delta t, \quad x_{n+1} = x_n + v_n\Delta t. \qquad (5) $$ These relations imply that $a_n$ and $v_n$ can be considered constant during sufficiently short intervals from the point of view of the change in velocity and position, respectively.
The numerical monitoring of general motions goes thus as follows: You start with some initial condition $x_0$, $v_0$, determine the initial acceleration $a_0$ based on the force law, and obtain from the iterations both $v_1$ and $x_1$. From these and the force law you obtain $a_1$, and the iteration can go on without any limit. Equation (5) expresses the essence of Newton’s law: if we know the force law/acceleration law to be valid at any time, out of a single rate of change, the acceleration, the full motion follows.
Vertical projection revisited as a general motion (by means of a hand written table)
Body on a spring. In this case the force is proportional to displacement $x$, $F =-Dx$, out of which the acceleration of a mass $m$ fixed to the spring is $$ a=-(D/m) x.$$ For the purpose of simplicity, $D/m=1$ is chosen. Follow the motion with iteration (5) starting from the initial position $x_0=1$ without any velocity: $v_0=0$ $(\Delta t=0.1)$