The plastic ball passes through the highest point of its trajectory
at about t=1.6 s.
At the instant that it passes through its highest position
the plastic ball has vanishing speed.
Shortly before and shortly after that instant of time
its speed is small.
We had observed that the deceleration of the plastic ball
was almost equal to the gravitational acceleration
shortly before and shortly after that instant of time.
Now, at low speed the plastic ball does not notice the presence of air.
But, at higher speed it does.
Hence, a good candidate for an explanation of the phenomenon
which we have observed here,
is the resistance against movement which is experienced
by objects which are submerged in air.
When you move fast, you feel
a much larger
air resistance
than at low speed.
Just stick your hand out of the window of a moving car
in order to observe the difference.
When the plastic ball moves upward it has initially a large velocity,
of about 70 km/h.
At such velocity, the air resistance experienced by the ball
is large and works against the upward movement
in the same direction as the gravitational force.
Hence, the deceleration is larger than without air resistance.
We indeed observed that the decrease in average velocity
was much larger for each interval at early times
than the about 1 m/s
that one expects from just the gravitational force.
When the plastic ball moves downward it gains velocity
because of the gravitational force.
But, now the air resistance works upward
against the gravitational force.
Hence, the downward acceleration is smaller than without air resistance.
We indeed observed that the decrease in average velocity
was much smaller for each interval at later times
than the about 1 m/s
that one expects from just the gravitational force.
Hence, we will assume here that the extra deceleration
is due to air resistance.
Now, we would like to have some idea about the relation
between the extra deceleration and the velocity of the plastic ball.
For that we must determine the instantaneous velocity
of the plastic ball at each instant of time.
Although the use of an accurate graph would be preferable,
for now we take for the velocity at a certain instant of time
v(t)
the average speed of the two adjacent intervals.
Moreover, we determine the extra deceleration at a certain instant of time
aextra(t)
by subtracting the about
10 m/s2
that one expects from just the gravitational acceleration,
from the decrease in velocity in the two adjacent intervals,
divided by the time interval in which this decrease is observed.
So, for the velocity at 0.1 s
we average the velocities of the first and the second interval,
to obtain
(19.0+17.2)/2 m/s = 18.1 m/s,
whereas the extra deceleration at 0.1 s
follows from
-10+(19.0-17.2)/0.1 m/s2
= 8 m/s2.
We do the same for other instants of time,
but we do not consider those instants of time
at which the extra deceleration is too small
and where all the errors in our not very clean method
become too large.
Below, we collect the results in a table.
instant (s)
velocity (m/s)
modulus of extra deceleration (m/s2)
0.1
18.1
8
0.2
16.4
6
0.3
14.8
6
0.4
13.3
4
0.5
11.9
4
0.6
10.6
2
0.7
9.4
2
0.8
8.2
2
0.9
7.1
1
2.7
9.9
3
2.8
10.6
2
2.9
11.3
4
3.0
11.9
4
3.1
12.5
4
3.2
13.1
5
Next, in order to figure out
how the velocity v
relates to the extra deceleration
aextra,
we assume three possible relations:
A linear relation of the form
aextra = A×v,
a quadratic relation
aextra = B×v2
and a cubic relation
aextra = C×v3.
In the table below we determine
A,
B
and C
at the instants of time shown in the above table.
instant (s)
A (s-1)
B (m-1)
C (s/m2)
0.1
0.44
0.024
0.0013
0.2
0.37
0.023
0.0014
0.3
0.41
0.028
0.0019
0.4
0.30
0.023
0.0017
0.5
0.34
0.029
0.0024
0.6
0.19
0.018
0.0017
0.7
0.21
0.022
0.0024
0.8
0.24
0.029
0.0035
0.9
0.14
0.020
0.0028
2.7
0.30
0.030
0.0030
2.8
0.19
0.018
0.0017
2.9
0.35
0.031
0.0028
3.0
0.34
0.029
0.0024
3.1
0.32
0.026
0.0021
3.2
0.38
0.029
0.0022
average
0.30±0.09
0.025±0.004
0.0022±0.0006
The average values for
A,
B
and C
are in the last line of the table.
We find that the variation in the values for
A
is about 30 percent,
in the values for B
about 16
and in the values for C
about 27 percent.
It gives a hint that the relation between the two quantities is quadratic.
But, a better method would be to make graphs
and deduce more precise values for the instantaneous velocity
and acceleration.