What factors will affect the time of a falling paper cone

The issue raised is the motion of paper made cones within air as they are released. The time that the cone will take to reach ground level from the moment of its release depends on several variables, which have to be investigated in order to conclude whether changing them would be relevant in affecting the period of time needed.

Hypothesis

There are several factors which theoretically could affect the time, assuming that the air is homogenous throughout the room and by using the same paper. We have to understand that a higher or a lower top speed will be the factor which changes the time needed. Thus the factors affecting top speed will have to be investigated. These are shape, top angle, height, type of paper and resistive forces. Some can be cancelled out before experimentation as we know from theory that a mass difference will not affect the free falling object (it will actually but to a negligible extent).

Experimentation

To investigate how different factors change the time needed one would have to set up a laboratory experiment. Immediately one encounters the first issue: how far will the paper cone will have to be put in order to reach its terminal velocity? As paper cones are quite light in terms of mass one can assume that a 3 meter chute is all right, as according to Newtonian physics the drag (air resistance), i.e. the upwards force will soon be equal to the weight of the paper. The aforementioned variables (top angle, resistive forces…) will have to be tested by using different values. To achieve this, the experimenter will have to probe those relationships.

Size: cones of different sizes but same paper will have different masses but this will not affect the time as weight is not a criterion after terminal velocity has been reached.

Drag coefficient: the drag coefficient is a dimensionless quantity which determines the aerodynamic properties of an object. The smaller it is, the lower the resistive values of fluid air. For instance for a normal cone it is of 0.5 whilst a cube has a drag coefficient (Cd) of about 1.05[1]. The formula to find this value is Cd= Fd / 0.5Vρ. The airs density is an uncontrollable constant whereas the mass shall be manipulated in order to see the effect. The experimenter will create three cones of different mass, let them drop off from a same height and observe. The time shall be measured.

Apparatus

Only simple tools such as a scale, a stopwatch or rulers shall be used, no complicated machines such as lasers or position sensors.

Variables:

Uncontrollable constants: air’s fluidity

Controllable variables: mass, shape, aerodynamics

Uncontrollable variables: time

Diagram of set up

CONES

Cone a is the sample cone

Cone b is the same shape and material as A but is an scaled up version. This is done so that the effects of mass difference can be investigated.

Cone c is the same size as A but is made of a different kind of paper, a rougher one. This is done so that the effects of the cone’s aerodynamics can be investigated.

Cone d is of different shape than the B but has the same surface area (thus the same mass). This is done to investigate how much the steepness of the sides will affect the cone.

The cone is dropped from 3 meters and simultaneously the stopwatch sets off. As it hits the earth the chronometer will have to be stopped. The figures are recorded. The process is repeated at least 5 times with all cones and thus an average is drawn. Thereafter the results are compared. The smaller the time needed, the more aerodynamic the shape is.

[1] http://web.archive.org/web/20070715171817/http://aerodyn.org/Drag/tables.html