Home:

Physics and Astronomy


Free Fall without and with Air Resistance



mass diameter
Enter mass (kg), and diameter (m) of the body,
mass density
or enter mass (kg) and density (kg/m^3),
e.g. 7874 kg/m^3 for iron,
diameter will be computed
density diameter
or enter density (kg/m^3), and diameter (m),
e.g. 7874 kg/m^3 for iron,
mass will be computed
data input
Enter acceleration (m/s^2), density of the air (kg/m^3), drag coefficient (dimensionless),
for a sphere the dag coefficient is cD = 0.4 .. 0.5 (0.47)
apply input enter maximum time (s), or Height (m),
and press button "Apply Input"

You may also use the Parachute mode:
parachute
time height
Enter time (s), or Height (m),
and press button "Apply Input".
apply input Using mass = 150 kg, diameter = 9 m,
drag coefficient cD = 1.3,
dome-shaped chute cD = 1.3 .. 1.5,
parasheet (flat sheet) cD = 0.75
the rate of descent will be about 5 m/s.
Using a "Height" input the delay due to air resistance is computed (∆t absolute, and ∆t/T in relation to the time T of free fall).



Theory

The equations without air resistance are quite simple. We assume g=const, s(t=0)=0,  and v(t=0)=0:

s(t) v(t)

Taking into account the air resistance force (drag force):
air resistance
                                force drag force
which depends on the drag coefficient cD, the velocity v, the density of air ρ, and the cross section A of the body,

the equation of motion is
formula ma
The air resistance force will increase until it equals the gravitational force m.g, and the body will then move at a constant velocity vlimit as the net force is m.a=0
limit
                                    speed velocity
The differential equation of motion is solved by
solution
                                differential equation

The drag coefficient cD is a function of the non-dimensional Reynolds number Re, defined by
Reynolds
                                number Re
L is a characteristic linear dimension (e.g. diameter of the body/sphere), ρ the density, and μ the dynamic viscosity (for air 17.9 ÁPa.s at 10░C).

>>In theory, the flow is laminar when Reynolds number is below 4,000. However, in practice, turbulence is not effective when Reynolds number is below 200,000; so when the Reynolds number is less than 200,000, you may assume laminar flow. In addition, when the Reynolds number is higher than 2,000,000, you may assume turbulent flow.<< (*).

Using a "Height" input the delay due to air resistance is computed (∆t absolute, and ∆t/T in relation to the time T of free fall).

Ferdinand Reich's experiments were performed at a height of 158.5 m using spheres of 4,034 cm diameter and 270,45 g mass.
The speed limit 86.3 m/s is not reached. The effect of air resistance on time is ∆T = 5.8676s - 5.6845s = 0.1831s, ∆T/T = 3.22%.

Luftreibung


Select "∆t - H":

air resistance time difference

The diagram below is showing the time difference as a function of diameter due to air resistance for iron balls (ρ=7874 kg/m^3) falling from 150 m:
Iron Ball air resistance

∆t/T ~ 1/diameter  (R2=0.999996)

The diagram below is showing the time difference as a function of mass due to air resistance for iron balls (ρ=7874 kg/m^3) falling from 150 m:

air resistance
                                mass

∆t/T ~ 1/mass  (R2=0.999992)



Parachute


The diagram is showing the rate of descent (vLimit, terminal velocity) as a function of mass, and the equivalent height at free fall.

parachute


Web Links
Freier Fall mit und ohne Luftwiderstand (virtualmaxim)

https://www.grc.nasa.gov/www/k-12/airplane/falling.html

(*)
Drag Force and Drag Coefficient


2016 J. Giesen

letzte ─nderung: 6. Nov 2016

mail
s