Download Kinematics and Air Resistance

Kinematics and Air Resistance
1
Object
To look at kinematics in one dimension and to study the effects of air resistance on falling objects.
To measure the terminal velocity of a falling object and verify how the terminal speed depends on
physical properties of the system.
2
Apparatus
Electronic Timer, Stopwatch, Measuring Tape, Meter stick, Steel Ball, Coffee Filters, Mass Balance.
3
Theory
We know if an object is freely falling toward the ground, subject to no forces other than gravity,
that it will fall with a constant acceleration directed downward equal to the acceleration of gravity:
9.8 m/s2 . The one-dimensional kinematic equations, assuming constant acceleration, describe the
relationships between position, velocity, acceleration and time for this object.
1
x = xo + vo t + at2
2
(1)
What happens if we also account for the presence of a non-negligible air resistance force? When
an object moves through air, it collides with the individual air molecules. These many, microscopic
collisions result in a resistive force opposing the motion of the object relative to the air. If the air
is at rest, this resistive force would then be in the opposite direction to the object’s motion, which
is true for an object falling toward the ground. The magnitude of this air resistance force depends
on the size and shape of the object, the density of the air, and the speed of the object. In general,
the magnitude of the air resistance force acting on such an object can be written as
|Fair | = b1 v + b2 v 2
(2)
where b1 and b2 are constants and v is the speed of the object. Usually, one of these terms is much
larger than the other. For very small diameter objects, like oil drops or raindrops, the first term
involving v 1 power dominates. For large objects, like cars or bikes, the second term dominates and
the first term is negligible. As a result, one could describe the magnitude of the air resistance force
as
|Fair | = bv n
(3)
where b is a constant called the air resistance force and n is an integer. For small objects, n = 1,
and for large objects n = 2.
Consider an object that starts from rest and falls toward the ground through the air. At first,
it is moving rather slowly so the air resistance force, which is proportional to v, is negligibly small.
The object is nearly in free fall. As the speed increases, however, so does the magnitude of the air
resistance force. This air resistance force would be directed upward, in the opposite direction to
the motion, and hence would oppose the gravitational force on the object. Newton’s 2 nd law would
allow us to determine the acceleration as
a=
(mg − Fair )
m
1
(4)
where a is the magnitude of the acceleration and m is the mass of the object. Notice if the air
resistance force can be ignored, the acceleration is that of gravity. As the object’s speed increases,
the air resistance force gets larger and the resulting acceleration gets smaller. The object may still
be speeding up, but at a slower rate. As the speed continues to increase, the magnitude of the
acceleration will continue to decrease. At some point, the speed will approach a value such that the
air resistance force (UP) is equal in magnitude to the gravitational force (DOWN). At this point,
the net force would be zero, there would be no acceleration, and the object would then move at a
constant velocity. This constant velocity is called the terminal velocity of the falling object.
n
mg = bvterm
(5)
would define the terminal velocity of the falling object if we knew m, b, and n.
4
Procedure
4.1
Part A. Falling Ball
1. Orient the Electronic Free fall Timer setup so that the ball will be released, fall some distance
d, and then strike the stop pad. Measure and record this distance d that the ball will fall. Use
the timer setup to measure the amount of time it takes the ball to fall this distance, starting
from rest. Repeat this time measurement 2 more times for a total of 3 time measurements.
2. Repeat this process for 9 other values of d. These values should be uniformly spread out
ranging from 2.0 meters or more down to 20 cm.
3. Measure and record the mass of the steel ball you used.
4.2
Part B. Falling Filter.
1. Now repeat the same basic process using a coffee filter. Rather than using the electronic timer,
you will just use a stopwatch and manually time the fall of the filter. This time take 5 time
measurements at each drop distance d. Measure and record d and the 5 time measurements.
The 10 drop distances should be rather uniformly spread out between 20 cm and 2.0 m or
more, just as with the steel ball.
2. Measure and record the mass of the coffee filter.
4.3
Part C. Varying Filter Mass - Terminal Velocity.
1. For this part you will be using a motion detector interfaced to the computer. Start the Data
Studio program and load in the activity called termspeed.ds. Place the motion detector on
the ground, facing upward. The space above the motion detector should be clear of objects
so that the detector, when working, simply “sees” the ceiling staying still.
2. Hold a single coffee filter directly above the motion detector, as high up as possible, i.e., at
the ceiling. As a test, drop the filter and see where it lands. You would like it to land right
on top of the motion detector. You might mark a spot on the ceiling with some tape from
where releasing the filter will result in it landing on the motion sensor. Activate the motion
detector and release the filter so that the detector will record the motion of the filter as it
falls toward the detector.
2
3. From your graphs, determine the terminal speed for the falling filter. This may be done by
looking at the position graph at the end of the fall and doing a linear fit there to get v. You
may also look at the velocity graph and average the speeds toward the end of the motion
where they should be fairly constant. You should print a copy of the position, velocity, and
acceleration graphs as a function of time for 1 dropped filter.
4. Now place a 2nd coffee filter inside of the first. The combination should now have twice
the mass but still have the same physical shape properties. Again, measure and record the
terminal speed for this falling combination of filters.
5. Repeat for 3, 4, 5, 6, 7, 8, 9 and 10 filters together.
5
Calculations
5.1
Part A & B. Falling Ball and Filter.
1. Calculate the average time the ball took to fall each of the 10 distances d for which you
took data. This should amount to averaging the three times you measured. You now have
10 different sets of time and corresponding position data. Make a graph of Position vs time
using this data.
2. Make a graph of Position vs time2 .
3. Calculate the average velocity for the ball during each of the 9 intervals defined by your
position and time data. Let’s assume that the average velocity of the ball during any interval
is approximately equal to the instantaneous velocity of the ball at the middle of the interval
time-wise. Calculate this center time for each interval and make a graph of velocity vs time.
~vavg =
(~xn+1 − ~xn )
(tn+1 − tn )
and
tcenter =
(tn+1 + tn )
2
(6)
4. Your 9 calculated velocities and corresponding times now define 8 intervals over which you
can calculate the average acceleration of the ball. Do this and calculate the center time of
each of these 8 intervals. Using these corresponding acceleration and center time values, make
a graph of acceleration vs time for motion of the steel ball.
~aavg =
(~vn+1 − ~vn )
(tn+1 − tn )
(7)
5. Your velocity graph may look rather linear. Do a linear regression to get the best fit line to
the data. Compare the slope of this line with the value you would expect if air resistance
is negligible. What does the y-intercept of this graph represent? Also do a linear regression
for your acceleration graph. Compare the slope and y-intercept with expected values. Either
your Position vs time graph or your Position vs time 2 graph should look linear. For whichever
one does, do a linear regression and compare the slope with the expected value.
6. Repeat steps 1-3 for your falling data with the coffee filter. After some point, does your
position vs time graph look fairly linear? Do a linear regression using only the data points
after the graph starts to look linear. Compare the slope of this graph with the appropriate
result from your data of part C. Also do a linear regression for the corresponding portion
of your velocity graph. Compare the slope and y-intercept of this graph with the expected
values.
3
5.2
Part C. Varying Number of Filters
1. You want to determine whether the air resistance force is proportional to v, v 2 , or v n . As
you may recall from the first lab, a handy way to determine this relationship is to construct
a graph of ln(m) vs ln(vterm) for your data, do a linear regression, and look at the slope
of that graph to determine n. Do this. Remember, m would just be the number of filters
dropped times the mass of one filter.
n
2. Now construct a graph of mass vs vterm
, where n is the integer you have determined from
step 1, using your data from Part C. Do a linear regression. Use the slope to determine
the coefficient of air resistance for the filters. Compare the y-intercept of this line with the
expected value.
6
Questions
1. How do your graphs using stopwatch data for the falling filter compare with those that you
recorded using the motion detector? Do they match up well or not? Do you expect them to?
Why or why not?
2. If you had a single filter that was the same mass but twice the cross sectional area of the
ones you used for this lab, would you expect it’s terminal speed to be smaller, larger or the
same as the one you used? What if it was twice the mass and twice the cross sectional area?
Explain.
3. How much does air resistance affect the motion of the steel ball as it falls? Defend your
answer using the data and results from this lab.
4
KINEMATICS/AIR RESISTANCE DATA SHEET
Part A - Falling ball
Trial
Units
Distance
Time 1
Time 2
1
2
3
4
5
6
7
8
9
10
Mass of Ball
5
Time 3
Part B - Falling Coffee Filter
Trial
Units
Distance
Time 1
Time 2
Time 3
1
2
3
4
5
6
7
8
9
10
Mass of Filter
Part C - Varying Filter Mass
# of Filters
vterm
m/s
1
2
3
4
5
6
7
8
9
10
6
Time 4
Time 5