Transcript falling ball presentation

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
Ruler
Vertical
Glass Tube
Data Collected
Time 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, V
 As the velocity at time t goes to the terminal velocity, the
acceleration goes to zero
 Replace 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:
v t
gt
V
1
gt
V
V
 We then integrate this equation to get
h t
tV
gt
V
V2
g
 This is the equation shows the distance fallen at any
time t
 The only value we do not know, is the terminal velocity
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 meters
 Gravity = 9.81 meters per second squared
 With these values in the equation, we solve for the
terminal velocity in Mathematica
gt
V
FindRoot h
t
V
V2
g
, V, 0.05
 By using FindRoot in Mathematica we find the
terminal velocity to be 2.22 meters per second.
 Now that the terminal velocity is known, we have the
model equation
h t
t 2.22
9.81 t
2.22
2.22
9.81
to show the distance fallen at any time t.
2
Plot this model equation as
well as the recorded data
points
Model Graph
h t
Air Resistance
Proportional
to Speed
3
2.5
2
1.5
1
0.5
t
0.2
0.4
0.6
0.8
1
1.2
1.4
Recorded Data Points Plotted
against the Model Graph
h t
Model
data
vs Recorded
Data
3
2.5
2
1.5
1
0.5
t
0.2
0.4
0.6
0.8
1
1.2
1.4
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.