Abstract
In this experiment, two different methods were used to obtain the acceleration of a free
falling object, gravitational acceleration. First, a weight was attached to a long paper tape,
which was put through the PASCO tape timer. As the weight falls, the tape timer marked
a dot on the paper every 1/40 s. For the second method, a glider was put on an air track,
and a mass was attached to it by a string. The string was then put on a pulley, at the end
of the track, and the mass was dropped. The photogate took the time that the glider
blocked at two different places along the track. The mass on the glider and the hanging
mass were varied five times. For both methods, the velocities were calculated, from
which the acceleration of the falling objects were calculated. The resulting accelerations
were 5.0m/s2 and 9.5m/s2 for the first and second method, respectively. Both results are
less than the literature acceleration of falling object, 9.8m/s2. This implies that there were
other forces that were acting on the objects that caused to have less acceleration. The
other forces could have been friction force or air resistance.
Data Analysis and Results
Time
(s)
0.000 ±0.001
0.050 ±0.001
0.100 ±0.001
0.150 ±0.001
0.200 ±0.001
0.250 ±0.001
0.300 ±0.001
0.350 ±0.001
0.400 ±0.001
Position
(mm)
15.5 ±0.5
33.2 ±0.5
62.7 ±0.5
103.7 ±0.5
157.1 ±0.5
222.8 ±0.5
300.9 ±0.5
391.0 ±0.5
493.6 ±0.5
Velocity
(m/s)
0.00
0.35 ±0.04
0.59 ±0.03
0.82 ±0.03
1.07 ±0.02
1.31 ±0.02
1.56 ±0.02
1.80 ±0.02
2.05 ±0.02
Average
Acceleration (m/s2)
Standard Error
Acceleration
(m/s2)
4.7 ±0.2
4.6 ±0.2
5.0 ±0.1
4.9 ±0.1
5.0 ±0.1
4.8 ±0.1
5.0 ±0.1
4.9 ±0.1
0.2
Table 1. The positions of the dots on the paper tape at every 2/40 s, and the caculated
velocity and the acceleration of the falling object. The positions measured by the meter
stick starting at 10mm. The calculated average acceleration is 4.9 ±0.1m/s2.
Figure 1. Velocity vs. Time graph drawn by hand. The slope of the line is 4.83, and the
y-intercept is 0.10 giving the equation y= 4.83x+0.10.
Velocity vs. Time
2.5
y = 5.00 (±0.02)x + 0.1(±0.1)
Velocity (m/s)
2
2
R = 0.9982
1.5
1
0.5
0
0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 2. Graphical analysis of Velocity vs. Time. The slope of the line is 5.00 ±0.02
m/s2, which is the acceleration. The R2 value of 0.9982 confirms the linear relationship
between the velocity and the time.
The acceleration of the falling object is obtained in three different ways. First, the
acceleration was obtained analytically, which gave 4.9 m/s2 as result. Other two methods
are graphing method. The first graph was drawn by hand velocity vs. time. This gave
4.83 m/s2 as the acceleration of the falling object. The slope of the line is the acceleration
because acceleration is the rate of change in velocity. The second graph is computergenerated, which give 5.00±0.02 m/s2 as the acceleration.
When the two graphs are compared, the accelerations are similar in value, and the
y-intercepts are the same; however, the computer-generated graph gives better value
(acceleration being closer to 9.8 m/s2). When the computer-generated acceleration is
compare to the value obtained analytically, 4.9 ±0.1 m/s2, they are very similar in value.
The percent difference is 2.0%. However, both values are very far from the theoretical
value of 9.8 m/s2, which probably resulted from the friction between the paper tape and
the PASCO timer.
Initial
Time (s)
0.206 ±0.005
0.166 ±0.005
0.124 ±0.005
0.131 ±0.005
Final
Time (s)
0.098 ±0.005
0.074 ±0.005
0.054 ±0.005
0.049 ±0.005
Initial
Velocity
0.48 ±0.06
0.60 ±0.06
0.80 ±0.06
0.76 ±0.06
Final
Velocity (m/s)
1.0 ±0.1
1.3 ±0.1
1.8 ±0.1
2.0 ±0.1
Acceleration
(m/s2)
0.71 ±0.1
1.280 ±0.08
2.432 ±0.04
3.135 ±0.03
Table 2. Times the photogates were blocked by the glider and calculated velocity and
acceleration of the glider and hanging mass. The length of the wing that blocked the
photogate is 0.099 ±0.005 m. The distance that the glider traveled is 0.560 ±0.001 m
M2
M
1
T1
M
1
FN
T2
M2
m1g
m2g
Figure 3. The set up and the free body diagram of the glider and the hanging mass. M2 is
the glider and M1 is the hanging mass
M1
T1- m1g =-m1a
M2
FN=m2g = 0
T2 = m2a
T1=T2
The acceleration for the both masses is the same because they move together
-m1a + m1g = T1 = T2 = m2a
m1 g = m2 a + m 1 a
m1g = a(m2 + m1)
m1 g = F
(1)
(2)
(3)
If m1g is plotted against the acceleration (a), then the slope of the line should equal to m2
+ m1, which is 314.9g. The plot of Force vs. Acceleration is shown below.
Acceleration (m/s2)
0.71 ±0.1
1.280 ±0.08
2.432 ±0.04
3.135 ±0.03
F (m1g) (N)
245 ±0.1
441 ±0.1
833 ±0.1
1029 ±0.1
Table 3. Calculated acceleration from Table 2 and the force which is m1g, where m1 is
mass of the hanging mass and g is the gravitational acceleration.
Force vs. Acceleration
1200
y = 326±9x + 22±2
1000
2
Force (N)
R = 0.9984
800
600
400
200
0
0
0.5
1
1.5
2
2.5
3
3.5
2
Acceleration (m/s )
Figure 4. Force vs. Acceleration. The equation of the line is y=326 ±9x+ 22±2, where y
is force, m1g, and x is the acceleration. The slope of the line is m2 + m1. The true m2 +
m1 value is 314.9g.
Rearranging equation (3) gives
m1
(4)
ga
(m1  m 2 )
m1
If acceleration is plotted against
, then the slope of the line is the gravitational
(m1  m 2 )
acceleration.
mass of the glider (g)
289.9 ±0.1
269.9 ±0.1
229.9 ±0.1
209.0 ±0.1
Hanging mass (g)
25
45
85
105
m1/(m1+m2)
0.079 ±0.004
0.143 ±0.002
0.270 ±0.001
0.333 ±0.001
Acceleration (m/s2)
0.71 ±0.1
1.280 ±0.08
2.432 ±0.04
3.135 ±0.03
Table 4. The mass of the glider with weights, mass of the hanging mass, hanging mass
divided by the total mass and the calculated acceleration from the Table 2.
Acceleration vs. m1/Total Mass
Acceleration (m/s 2)
3.5
y = 9.5 ±0.3x - 0.07 ±0.06
R2 = 0.9984
3
2.5
2
1.5
1
0.5
0
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
m 1/(m 1+m 2)
Figure 5. Acceleration vs. Hanging mass / Total mass. The R2 value clearly indicates the
linear relationship between the two. Also, the slope of the line is the gravitational
acceleration, which is 9.5 ±0.3 in this case.
Sample Calculations
Finding velocity and acceleration
Part I
x(change in position) 0.0332  0.0155
m
velocity 

 0.35
2
t (change in time)
s
0
40
v(change in velocit y) 0.59  0.35
m
Acceleration 

 4.8 2
4
6
t (change in time)
s

40 40
Part II
Initial velocity 
x(length of the glider) 0.099m
m

 0.48
t (Initial time)
0.206s
s
Final velocity 
x(length of the glider) 0.099m
m

 1.0
t (final time)
0.098s
s
V2 = V02 + 2a(d)
d=distance that the glider traveled (0.56m)
V 2  V02 1.0 2  0.48 2
m
a=

 0.71 2
2d
2(0.56)
s
Discussion
This experiment investigated the acceleration of free falling object by first, using
a paper tape and PASCO timer and second, glider on an air track. The experimentally
obtained acceleration for the first part was 4.9±0.1m/s2, and for the second part was 9.5
±0.3 m/s2.
Both of the accelerations are smaller than the literature value of 9.8m/s2,
especially the acceleration obtained by the first part of the experiment. This could have
resulted from the friction between the PASCO timer and the paper tape as the paper
moves through the timer. There is friction between the paper tape and the carbon paper
disk and between the hammer and the paper tape, which would have slowed the velocity
of the paper, therefore, resulting in less acceleration. Also, there could have been air
resistance that slowed the falling of the weight. However, since the weight was dropped
edge facing down, the air resistant would not have influenced the acceleration greatly.
In contrast, the air resistance could have been a major factor in the second part of
the experiment. The acceleration obtained in this part is much better than the
acceleration obtained from the first part. However, it is still less than the literature value.
This is because when the hanging mass was dropped; the flat side of the weights were
facing down, having more air resistance working against it. Therefore, dropping the
weight edge side down could give a better result. Also, the friction between the pulley
and the string could have been a factor, slowing the glider down, therefore, less
acceleration. Furthermore, the mass of the string was thought to be negligible, but it
would have had some effects in the acceleration.
The acceleration of free falling object was not obtained directly from the second
part of the experiment. Instead, the acceleration of the glider was obtained first by using
the equation V2 = V02 + 2ad. The term “d” is the distance that the glider traveled between
the two photogates. The initial and final velocities were obtained by using the time that
the glider blocked the photogates and the length of the glider. The calculations are shown
in the sample calculation for the glider with different masses and different masses
hanging on it. Then taking the accelerations of the glider, a graph can be plotted
m1
according the equation
g  a . This equation is derived in equations (1)
(m1  m 2 )
m1
through (4) in the Data Analysis. When the acceleration is plotted against
,
(m1  m 2 )
then the slop of the line is the gravitational acceleration, which is the acceleration of the
free falling object. The plot and the free-body diagrams are show in the Figures 3 and 5.
This experiment demonstrated that all the free falling objects have an acceleration
of 9.8 m/s2. Also, when two objects are connected, the acceleration of two objects are the
same regardless of their direction since they move together.
DISCUSS THE CHANGE IN VELOCITY CHANGE IN TIME THING!