Project H

Set Up

In a laboratory a long ruler is placed in a vertical glass tube. A stroboscope which flashes a 1000 times per minute is set up in front of the tube. A ball is released from rest at the top of the tube in a darkened room. As the ball is released, the stroboscope starts flashing and the flight of the ball is recorded by a camera with an open lens. The result is a photo on which the position (relative to the ruler) of the ball at fixed time instants is recorded.

Falling Ball

Stroboscope Flashes 1000 times per minute

Vertical Glass Tube

Ruler

Data CollectedTime in seconds

Distance in meters

0.00 0.00

0.09 0.04

0.15 0.10

0.21 0.20

0.27 0.33

0.33 0.50

0.59 0.70

0.45 0.93

0.51 1.19

0.57 1.47

0.63 1.78

0.67 2.10

Assumption

Assume that the magnitude of the force of air resistance is directly proportional to the speed.

P[v[t]]=α*v[t]

Construct a model to obtain the distance the ball has

fallen as a function of time, t

Start with Newton’s Second Law

m*a=F

where: a=v`[t] and Force is equal to the force acting

on the ball minus the resistance due to air

F= F-P[v[t]]=F-α*v[t]

m*v`[t]= F- α*v[t]

Divide through by the mass of the ball

v`[t]=g-k*v[t]

Terminal velocity, VAs the velocity at time t goes to the terminal

velocity, the acceleration goes to zeroReplace k with the terminal velocity, with the

known equation:g=k*V

Through manipulation we end up with the differential equation:

v`[t]=g(V-v[t]) V

We solve this equation by using DSolve in Mathematica to get:

vt gtV1 gtVV

We then integrate this equation to get

This is the equation shows the distance fallen at any time t

The only value we do not know, is the terminal velocity

ht tVgtV V2

g

Find the unknown parameter

In order to find this value we will use one of the data points from the previously collected data.

Time = 0.51 seconds Distance fallen = 1.19 metersGravity = 9.81 meters per second squared

With these values in the equation, we solve for the terminal velocity in Mathematica

By using FindRoot in Mathematica we find the terminal velocity to be 2.22 meters per second.

FindRooth t V

gtVV2g

,V, 0.05

Now that the terminal velocity is known, we have the model equation

to show the distance fallen at any time t.

ht t2.22 9.81t2.22 2.222

9.81

Plot this model equation as well as the recorded data

points

Model Graph

0.2 0.4 0.6 0.8 1 1.2 1.4t

0.5

1

1.5

2

2.5

3

ht Air Resistance Proportional to Speed

Recorded Data Points Plotted against the Model Graph

0.2 0.4 0.6 0.8 1 1.2 1.4t

0.5

1

1.5

2

2.5

3

ht Model data vs Recorded Data

Conclusion

The model found had the distance that the ball fell increasing at a slower rate than the data points recorded show.

This model therefore is probably not the best for implementation because the recorded data does not correspond very well to the model found.