Method of small steps

The method of small steps is a physical application of Euler's polygon method , which is used for the approximate mathematical description of movements.

If, for example, the acting force is not constant, then with simple mathematics it is not possible to evaluate Newton's first law because the acceleration is not constant. At the simplest level, the acceleration is assumed to be constant for a time interval.

Application example: near-earth free fall

The method of small steps is used, for example, when moving in free fall.

Physical background

• In the case of a free fall near the earth, the speed of a falling body would increase by 9.81 m / s per second if the air resistance was neglected. Then free fall would be an evenly accelerated movement . A parachutist who drops from a stationary balloon initially gets faster and faster, but his speed increases steadily. Its acceleration corresponds to the acceleration of gravity and is greater than that of a car: after one second it theoretically has a speed of v = 9.81 m / s (approx. 35 km / h), after two seconds 19.62 m / s ( approx. 71 km / h), after three seconds 29.43 m / s (approx. 106 km / h). In a real free fall , i. H. in a vacuum , the speed would continue to increase linearly accordingly. ${\displaystyle v}$
• In fact, however, the air resistance also affects the skydiver , which increases quadratically with the speed. The resulting acceleration therefore only corresponds to the acceleration due to gravity at the beginning, after which it decreases until after approx. 7 seconds the acceleration becomes zero - the parachutist now falls at the limit speed of the human body of approx. 55 m / s (approx. H). However, this speed is not the maximum speed, but that which is reached when taking the stable position perpendicular to the fall with arms and legs spread. In a straight, vertical posture head first, the air resistance is significantly lower and speeds of just over 500 km / h are reached.

Using a spreadsheet

With the help of a spreadsheet , however, such problems can be broken down into many simple and above all solvable subtasks, the results of which can be put together by the computer program to form an overall solution. The advantages are apparent:

• You don't need any knowledge of higher mathematics
• The integration is replaced by summing. The result is not exact, but satisfies most practical requirements.
• On the basis of intermediate results, you can immediately recognize small errors that can be corrected.
• The many verifiable interim results increase confidence in the result.
• By adding other relevant formulas, the solution can be gradually adapted to reality.

The procedure is always the same: With elementary formulas, relevant quantities such as force, acceleration or temperature are calculated for a certain point in time - these are the initial values ​​for the next point in time. The results are only correct if little changes from one point in time to the next . How big these changes and, above all, each time step can be, can easily be seen from the results. Complex formulas, such as those used in weather forecasts, cannot be evaluated any other way.

Individual formulas of free fall with air resistance

In the following calculation it is assumed that a spherical iron meteor with mass m  = 4 g and cross-sectional area A  = 1 cm²  enters the atmosphere at a speed of v = 15 km / s and is slowed down. We are looking for speed and braking deceleration as a function of altitude. These values ​​are used in known formulas and recalculated for each time step. The individual results are combined in the table to form the sizes you are looking for and then output graphically. The process is started at a sufficiently high altitude h where the air resistance is still negligible.

• The gravitational acceleration of the earth decreases with increasing distance h above the earth's surface. This is true

${\displaystyle a_{\mathrm {gravi} }=9{,}81\,{\frac {\mathrm {m} }{\mathrm {s} ^{2}}}\cdot \left({\frac {6370000\,\mathrm {m} }{6370000\,\mathrm {m} +h}}\right)^{2}}$

• At this altitude h the density of the air is only ( barometric altitude formula )

${\displaystyle \rho (h)=\rho ({\text{Boden}})\cdot e^{-{\frac {h}{8400\,\mathrm {m} }}}}$

• The flow resistance in air F _air at the velocity v also depends on this density

${\displaystyle F_{\text{Luft}}=0{,}5\cdot \rho (h)\cdot C_{w}\cdot A\cdot v^{2}}$

• With the flight direction to the center of the earth, the effective acceleration on the meteor of mass m is the difference between gravitational acceleration and braking acceleration

${\displaystyle a_{\mathrm {gesamt} }=a_{\mathrm {gravi} }-{\frac {F_{\mathrm {Luft} }}{m}}}$

• With this intermediate result, the then valid speed can be calculated one time step dt later

${\displaystyle v_{\mathrm {neu} }=v_{\mathrm {alt} }+a_{\mathrm {gesamt} }\cdot dt}$

• and from this the place where the meteor is then. This starts a new cycle.

${\displaystyle h_{\mathrm {neu} }=h_{\mathrm {alt} }+v_{\mathrm {neu} }\cdot dt}$

The calculation is carried out step by step with elementary means and corresponds to a simple integration which delivers usable results with a sufficiently small dt . Especially for the last two steps, there are better, but also more complex processes, which are described in Numerical Integration . Their use is often exaggerated if only a quick overview is required or - as in this example - the formula for the flow resistance for supersonic speeds does not apply exactly.

Numerical solution

Calculation table for free fall with air resistance

Deceleration of a meteor in the atmosphere

First, the parameters in cells J1 to J5 and the starting values ​​in A3, B3, C3 are specified; these values ​​are required almost everywhere in the table. In other programming languages ​​one would speak of “global variables”. The formulas just listed are programmed in adjacent columns of the spreadsheet, the intermediate results are usually processed in the columns further to the right. The “switching” to the next line is done by using the result of cell G3 to calculate the content of cell B4 according to the following time step. Finally you copy the formulas of the 3rd or 4th line into the next 2000 lines - the result is calculated at the same time.

Of decisive importance for the physical correctness of the results is the sensible choice of the time step dt , which should be as small as possible and in the table on the right has the very high value of 0.2 s - for this task. In the vicinity of cell G20, this leads to just acceptable value jumps of around 40%. However, even an increase to dt  = 1 s does not cause any serious changes, which demonstrates the robustness of this solution method.

In the picture opposite, the table shows the total acceleration as a function of the height. The surprising results:

• The meteors are slowed down hardest at a height of about 40 km, almost regardless of their mass, and can be broken up into fragments or burn up .
• The speeds in the last few kilometers above the earth's surface are always around 40 m / s - if the fragments have not burned up by then. The calculated speed curve is shown in the picture below.

Further investigations

The method described invites you to vary parameters such as size and initial speed and to examine their effects on the calculated results. This type of “ experimental math ” can lead to a greater understanding of the physics involved than examining the complex formulas in the previous paragraph.

Web links

Wikibooks: Corresponding section in "Information technology basics (ITG)"

• Method of small steps with constantly accelerated movement ( LEIFI )
• the small steps with vibrations ( LEIFI )