Slate litter

The initial conditions, i.e. starting location(x,y) and starting speed (magnitude and angle of v) are entered at the top left. The size of the sphere (radius in meters) and its density (in ^kg/m3) are entered in the field below. The physical effects to be taken into account in the calculation are selected by ticking the boxes. The calculation is started with the "Calculate trajectory" button. The trajectory is displayed after a few seconds. If the displayed section is not suitable, the image section can be selected automatically using the "Auto" button, or the image section can be varied over many orders of magnitude using the slider. The image section can be moved by pressing and holding the left mouse button. Various functions can be displayed in an extra window, allowing you to study the movement process in more detail. The physical quantities to be displayed as a function must be selected for the abscissa(x-axis) and the ordinate(y-axis). The window is opened by ticking the box at the bottom left. The trajectory is shown as a red line, the earth's surface as a black line. The coordinates of a point can be read off next to the mouse pointer. This makes it easy to determine the throwing distance, for example. The trajectory of the current curve can be saved by pressing the "Remember curve" button. If the "Show 2nd curve" checkbox is selected, this curve is displayed in gray as a comparison curve in addition to the curves calculated later with other parameters.

The trajectory is calculated using the 4th order Runge-Kutta method. The increment in time is set depending on the speed (^10-12sto ^10-2s). The coordinate system is selected so that the coordinate origin(x,y)=(0,0) lies on the earth's surface. The calculation of the trajectory is stopped when the earth's surface is reached or the number of 10,000,000 time steps has been exceeded. A constant acceleration due to gravity is fixed at 9.81 ^m/s2 and points in the negative y-direction. If it is location-dependent, it is calculated using the law of gravity as a vectorial quantity pointing to the center of the earth. It is assumed that the air resistance is proportional to the square of the speed, proportional to the density of the air and proportional to the cross-sectional area of the sphere. The cw value of the sphere is cw=1. The density of the air is calculated for the air resistance using the barometric height formula and is dependent on height. For Brownian motion, a normally distributed displacement of the particles is calculated using random numbers, which depends on the size of the particles, the viscosity of the air and the temperature. The model for this comes from Einstein (see Annalen der Physik. 17, 1905, pp. 549-560). Brownian motion assumes the properties of air at the earth's surface (20 °C, 1013 mbar). For the velocity distribution of the particle, a Maxwell-Boltzmann distribution is assumed at 293 K corresponding to the mass of the particle. To estimate the accelerations, it is assumed that the collisions of the air molecules with the particle have acceleration distances of a typical chemical bond length (assumption 0.2 nm). The buoyancy in the air can also be taken into account. The density of air at the earth's surface is 1.2 ^kg/m3. To calculate the Coriolis force, it is assumed that the Earth's axis is perpendicular to the image plane and that the Earth rotates anticlockwise. This corresponds to a viewing direction from north to south and a coordinate origin on the equator. The following were not taken into account: relativistic effects, changes in frictional properties at very low pressures, tidal forces, wind, etc.