Falling Objects

Purpose

Determine the differential equation of motion and its solution for a falling object

Theory and Discussion

Objects falling through the air are subjected to two forces-the force of gravity and air resistance.

The net force on the body = weight + drag (air resistance); By Newton's second law F=ma we can write the differential equation of motion:

my" = -mg - F_d where y is the height at time t, g is the gravitational constant, m is the mass of the object and F_d is the drag force caused for air resistance.

Through experimentation it has been found the the drag force is affected by several factors including the shape of the falling object, the viscosity (slipperiness) of the medium it is falling through and how fast it falls. For relatively slow objects, like a marble falling through syrup, we have laminar flow and the resistance is proportional to the velocity F_d = ßv, where ß is some constant of proportionality.

For objects falling relatively fast, we may have turbulent flow and the air resistance is higher, proportional to v²: F_d = ßv^2*

Using the fact that the drag depends on the shape of the object, the density of the medium it falls through, and its size it has been shown that:

C is the numerical drag coefficient, rho is the density of the medium, A is the cross-sectional area of the falling object, and v is the velocity.
In order to make the velocity positive, v>0, on the graph, we will make up to be positive and -mg indicates downward force. Since the drag must be in the opposite (upward) direction, it will be a positive term in the differential equation.

↑ > 0

The 2nd order differential equation of motion for the falling body, height y, velocity v, at time t , now becomes:

Part I Writing the differential equation of motion for your sphere

g = 9.8 m/sec²
C_sphere = 0.5 (dimensionless quantity)
Density of air at room temperature = 1.2 kg/m³

Measure the following for your sphere:
m =                            kg
A =                             m² (cross sectional area, not surface area)

Write your differential equation solved for y" =                                      (round term coefficients to two decimal places)
Write your initial conditions:
v(0) =                  m/s
y(0) =                  m Since you are experimentally dropping a ball (see below), use this height here - the higher the better.

Write your second order differential equation as two first order differential equations with dependent variables y and v.

(i)

(ii)

Using the Tool Phase Plane Plotter, yv version, do the following: http://www.butlercc.edu/mathematics/math_courses/ma260/MA260_resource_page.cfm

Plot y vs. t graph with scales

Use your graph to estimate the time it will take to hit the ground = sec

Plot v vs. t graph with scales

Use your graph to estimate the terminal velocity of the falling object = m/s

Gathering experimental data

Take a movie of a falling ball with your cellphone, digital camera, or other device.

Using a video editor such as Windows Movie Maker (included in Windows), iMovie, or one on your cellphone, find the position vs. time of the ball.

You will need the timer on the movie editor to find the time at each position of the ball:

Using your initial height, y₀ meters, at t₀ = 0 sec (whatever your timer reads at the top becomes 0 sec), measure down to each ball position in your movie. Using a ratio you can find the actual distance traveled. For example, if you drop the ball from 5 m, and your ruler measures 20 cm from the highest position to the ground, each cm is 5/20 = 0.25 m.

Data Table:

time (sec)	position (m)
t₀ = 0	y₀ = m

Plot these data on your calculator or computer and compare to the y vs. t graph above.

Time to hit the ground from your y vs. t graph sec Time to hit the ground from your video sec

* Technically it's F_d = -ßv|v|; when v<0 (dropping) F_dgoes up; when v>0 (rising), F_dgoes down