Abstract:

Presentation describes time dependent acceleration, velocity and distance with influence the air resistance. In schools it is impossible to practice researching free fall and moving objects with regard to air resistance. We have been solving this theoretically with numerical methods and computer simulation.

Free fall is a type of motion in which the only force acting upon an object is gravity. Objects that are said to be undergoing free fall, are not encountering a significant force of air resistance. They are falling only under the influence of gravity. Under such conditions, all objects will fall with the same rate of acceleration, regardless of their mass. A feather and stone will fall with equal velocity in a vacuum.

In presence of air, we have other conditions and therefore unequal velocities. It is easy to express these dependences with hyperbolic functions. However, in secondary and high schools these functions are unknown, very hard for use and to understand them. We found out other tools to solve this task. To avoid complex mathematical functions, we have been solving it with numerical methods.

By a falling object with air resistance, velocity is changing, and therefore air resistance force (ARF) is changing. Difficulty is, that ARF versus velocity is square dependent. After a sufficiently small period of time dt we have new velocity v1 and therefore new ARF. When an object falls through air, the ARF is increasing with square and therefore is decreasing an acceleration. Furthermore, after certain period of time, velocity reached terminal velocity vt.

After small time period dt we have another velocity v1 and thus another ARF. We used a small enough time interval dt and calculate new velocity v1. We did it with small intervals dt through arbitrarily long distance or sufficiently good approximation to reach terminal velocity vt. To solve this task, we used numerical simulations and computer algorithm. Acceleration, velocity and distance versus time dependences are presented with graphs.

With this methods we have been researching falling objects with different initial velocities vi (vi = 0, vi < vt, vi > vt, and vi = vt) and other parameters (density, drag coefficient,...).

Methods are helpful for moving projectile through the air with arbitrary initial velocity and slope. We have researched an inclined throw. Total velocity, acceleration and trajectory were found by adding the vertical component of the velocity and the horizontal component of the velocity. Velocity, acceleration and trajectory dependences versus time are presented with graphs.

Our methods are useful for researching falling objects, because it is impossible to do it in nature or in laboratories. Programs are based on Wolfram Mathematica.

Keywords:

Air resistance, terminal velocity, inclined throw, numerical simulations.