Influence of Airbags and Seatbelts during Head-on Car Collisions
IB Extended Essay – Physics
Name: Radhika Goyal
Candidate Number: 002762-017
Supervisor: Curtis Hendricks
Schools: American International School/Dhaka
Date: 5/12/2011
Word Count: 3,707 words
Radhika Goyal, 002762-017
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Abstract
The research question of this essay is “how does force felt by the passenger vary in
airbag/seatbelt versus non-airbag/seatbelt cases during head-on car collisions?” To
explore the question, a software called Logger Pro was used to analyze real-life car crash
videos. The videos were both in slow motion and real time, and this essay is divided into
two scenarios: one with seatbelt and airbag; and the second without. Logger pro helped
extricate the distance-time graphs of different objects in the frame such as the car itself
and the dummy’s head. It was crucial to properly treat the data by finding the slow
motion time factor to differentiate between real time and it’s equivalent in slow motion.
Using mathematics as a tool, I found the derivatives of the distance-time graph to find the
velocities and then the second derivative to find the accelerations with respect to time
during which the car crash occurred (please see graphs in section 4). Further more,
concepts from mechanics such as Newton’s laws of Motion were used to calculate the
force felt by the dummy by setting the impulse (the product of force and change in time)
equal to the momentum (the product of mass and change in velocity). The manufacturer’s
value for the dummy’s head’s mass was used (4.5 Kg). The uncertainty on the velocity
was designated to be the standard deviation of the real-time velocities at different time
intervals calculated for the car-dummy system prior to the collision. The results showed
that the presence of airbag and seatbelt made the force significantly lower due to
increased time of impact. Finally, possible sources of systematic and random error were
discussed (including parallax and interlacing of frames) and the limitations and ways to
improve the experiments were explained.
(Word count: 292)
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Table of Contents
1. Introduction
1.1 Research Question and Hypothesis
2. Analytical determination of the force using Newton’s Laws of
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5
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Motion
3. Planning the Investigation
4. Data Collection and Processing
4.1 Verifying the Speed before the collision and Designating
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Uncertainties
4.2 Calculation of the Slow Motion Time Factor
4.3 Results in Scenario 1: Without Seatbelt/Airbag
4.4 Results in Scenario 2: With Seatbelt and Airbag
5. Conclusion and Evaluation
6. Bibliography
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1. Introduction
In my essay, I want to address the physics behind collisions in car crashes so that I could
apply my knowledge in this subject to the real world. I focused on two particular
scenarios to gather data about a real life car crash: the first, without air bags or seatbelts;
and the second, including both of the said things.
In a world where almost 3,500 people die in car crashes everyday (World Health
Organization), research conducted on the application of seatbelts and airbags is relevant
to find out exactly what transpires in a car crash and how such devices can save our lives.
It would be interesting to investigate exactly how simple, yet ingenious, innovations such
as seatbelts and airbags come into play in real time a car crash to minimize the injury felt
by the passenger. I was curious to see how efficient they can be in comparison to a case
where the passenger is left completely unsecured.
Though it may seem daunting at first to gain enough raw data to draw valid conclusions, I
used an unorthodox way of collecting data of real life car crashes. Using a computer
software called Logger Pro and video editing software, Final Cut Pro, I was able to pinpoint key video clips from a moving called “Understanding Car Crashes” from which I
extricated distance-time graphs. From here, I could use my knowledge in mechanics and
Newton’s Law’s of Motion to calculate the Force felt that would be an indicator of the
efficacy of seatbelts and airbags.
Even though I was not in the laboratory or in an institute where professionals carry out
such car crashes on different car models, I would be able to gain first hand data that
would allow me to conclude whether or not applications of seatbelts and airbags can
make a substantial difference to the outcome of a car crash.
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1.1 Research Question and Hypothesis
To focus my essay, I addressed the two scenarios in my research question:
“How does force felt by the passenger vary in airbag/seatbelt versus non-airbag/seatbelt
cases during head-on car collisions?”
The force felt by the passenger and the acceleration experienced would allow me to
conclude how severe the crash was in the two cases, and hence the difference made by
the safety apparatus in the car.
I predict that the presence of the airbag and the seat belt play a vital role in extending the
time it takes for the passenger to come to a stop. By doing so, it probably changes the
force felt during that time as the dummy accelerates from the speed at which the car is
moving to a standstill.
2. Analytical determination of the force using Newton’s Laws of Motion
From our knowledge in mechanics, we know that Velocity =
DDistance x 2 - x1
=
DTime
t 2 - t1
(Young). Furthermore, this means that the slope (or the derivative) of a distance-time
graph is in fact the velocity of the object. In addition to that, it is also known that
Acceleration =
DVelocity
(again, the derivative of a velocity-time would imply the
DTime
acceleration). Both velocity and acceleration are vector quantities that have a horizontal
and vertical component, and for this essay, only the horizontal direction will be analyzed
since that is the direction of the car’s movement and the vertical component is relatively
constant.
The Logger Pro software would assist in the collection of the primary data in the form of
distance-time graphs. As the acceleration is sought, the second derivative of the distancetime graph would give us the required measurement.
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From there, the next step would be to use the reading of the acceleration in the equation
derived from Newton’s Second Law: Force = (mass)(acceleration) or its alternative
form (Force)(DTime) = (mass)(DVelocity) . The force will be compared in both
airbag/seatbelt and non-airbag/seatbelt cases to answer the research question at hand.
3. Planning the Investigation
Method:
1. Video samples from movies were chosen based on the control of variables such as
mass of the car; the clarity and quality of the video; and the visibility of the a
particular point.
2. Out of various video samples of crash tests, videos of two real crashes will be
analyzed. There are many different versions of these two scenarios (one of a
collision without an airbag and seatbelt and one with the two) including:
a. Real time or a slow motion run,
b. Moving or motionless camera,
c. Different distances from the car to the camera, which result in a change of
scale (size of the image in the video).
3. It is important to note that the two videos contain the same model of the car and
the same kind of dummy; and for this reason, variables (such as mass, size and
shape) will be sufficiently controlled and the comparisons of the force, impulse
and momentum can be valid when looking at the two scenarios.
4. Logger Pro was used to gather information on the distance, velocity and
acceleration of the object travelling at different points in time. This allowed the
analysis of different points in the video, for example, the movement of the
dummy’s head in comparison to the movement of the car.
5. At first, the initial speed of the car was tested in real time and from a video that
was recorded from a moving camera. From the commentary in the movie, it
would have been easy to know this initial speed as it was mentioned in the movie,
30 miles per hour, however, finding the speed from the video was a good way to
test the software and the validity of the data gathered from it. This test was carried
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out in Graphs 1, 2, and 3. Eventually, the standard deviation of the points
collected in this test served as the uncertainty on velocity for the remaining
analysis.
6. To analyze a video clip, it was imported into Logger Pro.
7. The scale inside the video was set by using the markings on the car. Professional
car crash testers have two standardized markings on the cars, which are exactly 2
feet or 24 inches apart. The distance between the two points was selected and
designated the said amount on the software.
8. Next, in order to select the points on the graph, it was necessary to sift through the
video frame by frame and select a particular point that remained clearly in the
view for as many frames as possible. In some videos, it was the circle enclosed
inside the tire of the car (sideways view) and in others, it was the circle on the
dashboard of the car. Either way, both of these points would show the motion of
only the car. As for the brain of the dummy, a pre-existing cross on the side of the
head was used to follow its path over time.
9. As the point collection began, it was necessary to follow the chosen point through
each of the consecutive frames by simply clicking on its new position each time.
Each position identified appears on the distance-time graph as a co-ordinate,
which can then be further analyzed with the application of linear fits and
statistical examinations.
The point that was “followed” in different time frames can be seen from the screenshot
below, the green line shows the scale marking (Picture 1):
Picture 1: Logger Pro
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4. Data Collection and Processing
4.1 Verifying the Speed before the collision and Designating Uncertainties
This is the first sample for this essay’s video analysis. In this sample, the video is shot
through a moving camera. As a result, it is important to take the speed of the camera into
account when trying to find the final velocity of the car before the impact. This is done by
taking a reference point inside the video itself that is assumed to be stationary and
marking its movement with respect to time.
The average velocities in the two relative motions are added and the result confirms the
given value of 30 mph from the video “Understanding Car Crashes” (the source of the
sample). The uncertainty on the velocity is designated to be the addition of the standard
deviations from the “forward” and the “backward” velocities (±0.663 mph). Please refer
to the calculation shown in Table 1 below. The precision on time was taken to be to two
decimal places, hence the uncertainty on that was ± 0.01 seconds.
Data Table 1: Real Time Car Moving Forward Real Time
Graph 1: Real Time Car Moving Forward Real Time
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Data Table 2: Real Time Camera Moving Real Time (Relatively Camera moves
backwards)
Graph 2: Real Time Camera moving backwards
Table 3: Calculating the Initial Speed from Diagram
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Data from Logger Pro
Real Time Car Forward
Time
Distance X
Velocity x
Distance Miles
Velocity x
Sec
Inches
In/sec
Miles
miles per hour
1
1.502
86.543
-131.469
0.00137
-7.471
2
1.543
81.101
-132.458
0.00128
-7.528
3
1.585
75.412
-132.458
0.00119
-7.528
4
1.627
70.217
-135.423
0.00111
-7.696
5
1.668
64.280
-141.552
0.00101
-8.044
6
1.710
58.096
-141.602
0.00092
-8.047
7
1.752
52.406
-139.377
0.00083
-7.921
Average
-7.747
Standard
Deviation*
± 0.252
Real Time Camera Backward
Time
Distance X
Velocity x
Distance Miles
Velocity x
Sec
Inches
In/sec
Miles
miles per hour
1
1.502
53.936
380.848
0.00085
21.644
2
1.543
69.990
376.393
0.00110
21.391
3
1.585
85.662
370.028
0.00135
21.029
4
1.627
100.188
376.138
0.00158
21.376
5
1.668
117.007
389.504
0.00185
22.136
Average
21.514
Standard
Deviation*
Total Speed
(mph)
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± 0.410
Total Uncertainty*
29.263 (mph)
± 0.663
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* The uncertainty on the measurements were determined by the variance of the data
which was calculated by the standard deviation of the points collected in an Excel
Spreadsheet. Please note that this uncertainty will be applied to all the measurements
in this essay’s video analysis (± 0.663 mph or its equivalent ± 11.675 inches/second in
real time).
Sample calculation of Velocity and conversion into miles per hour:
"distance
"Time
(81.101 # 86.543)inches
=
= #131.469inches /second
(1.543 #1.502)seconds
'
$ 3600seconds '$
1mile
= (#131.469inches /second)&
)&
)
% 1hours (% 63,385.7inches (
= #7.471milesperhour
TotalVelocity = VelocityCamera " VelocityCar
Velocity =
= 21.644mph " ("7.471mph)
!
= 29.115mph
Total Absolute Uncertainty on Velocity = StdDeviationCar + StdDeviationCamera
!
!
= 0.25286 + 0.410
= ± 0.633 mph (correct to 3 decimal places to match the precision of the velocity
!
measurement).
From this calculation, we see that the car was travelling at constant speed before the
crash. When standard deviation is taken into account, the given value of 29.263 miles per
hour is within the given range in the video of 30 mph. Hence, as the given value is
confirmed by the data, 30 mph is the speed that assumed for the rest of the analysis and to
calculate the slow motion time factor.
Graph 4: Slow Motion Car From Stationary camera: Distance Vs Time before Crash and
right afterwards. This graph is added to this essay to help with the calculation of slow
motion factor
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This graph indicates the slow motion constant speed at which the car was travelling
(shown by the linear fit in the selected shaded region), and then the curve due to
acceleration after colliding with the wall at approximately 33.55 seconds (±0.01 seconds).
4.2 Calculation of the Slow Motion Time Factor
Previously, it was shown that the car was travelling at approximately 30mph or 528
inches/second. In slow motion (graph 4), the same constant speed is shown to be 30.13
inches/second.
Distance travelled in the shaded blue region in graph 4 was from 44.14 – 29.09 = 15.04
inches. This should have taken the vehicle travelling at 30 mph:
Time =
Distance
=15.04 inches/528(inches/second) = 0.0285 sec
Velocity
However, instead, in slow motion, it took 33.53 – 33.03 = 0.50 seconds
Hence, to go from slow motion to real time, one must multiply 0.50 seconds/0.0285
seconds = 17.5 times.
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Real Time = Slow Motion *(0.0570) (This calculation will be used later to calculate the
final forces).
In slow motion, the time duration for twelve frames was 0.0285 seconds
Hence, there are 0.0024 seconds/frame or about 415 frames per second (fps).
4.3 Results in Scenario 1: Without Seatbelt/Airbag
The starting time for all the following graphs in scenario 1 is approximately 32.8 seconds.
The data collected for both the head and the car in no seat belt/airbag cases starts from
32.80 till about 36.00 seconds (this will be the range of the x-axis in Scenario 1 graphs).
From the distance-time graph (graph 5), we can see that the head accelerates from 34.99
seconds to 35.62 seconds. In the same time period, we can see from the velocity-time
graph (graph 6) that the velocity changed from -21.079 in/sec to 8.646 in/sec. The mass
of dummy’s head is 4.5 Kg (known from the manufacturers). It is important to note that
the velocity and time measurements are in slow motion hence they must be multiplied to
the time factor calculated in section 4.2. In addition, the value for Δ velocity must be
converted to the standard units, meters/second.
The change in slow motion velocity (in/sec) = 8.646 – (-21.079) = 29.725 in/sec
Ê 29.725in ˆÊ 1sec slowmotion ˆÊ 1meter ˆ
˜ = 13.25m /s
Á
˜Á
˜Á
Ë sec slowmotion ¯Ë 0.0570sec realtime ¯Ë 39.37inch ¯
The change in time in slow motion was 35.62 sec – 34.99 = 0.63 second = 0.036 seconds
(real time)
(Force)(DTime) = (mass)(DVelocity) (Newton’s impulse-momentum theorem)
(Force) =
(mass)(DVelocity)
(DTime)
Force = (4.5 Kg) (13.25 m/s) / 0.036 = 1660.4 Newtons
Using Force = mass* acceleration
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Average Acceleration = 1660.4/4.5 = 368.97 = 370 m/s/s (2 significant figures). This
number if roughly equivalent to 38 times the acceleration due to gravity.
This would be the average force felt by the dummy’s head if the force was kept constant
throughout the time of impact. However, we know that acceleration is not constant in that
time frame (evidenced by the acceleration-time graph), and the maximum force can be
calculated from the peak of Graph 7. From the Logger Pro software, we know that the
peak occurs at (34.98 sec, 103.7 in/s/s). After being converted to real time, the
acceleration would be approximately 810 meters / second/second (following similar
calculation as shown previously). From this measurement, the maximum calculation
would be around Fpeak = (4.5 Kg)(810.71m/s/s) = 3648.2 Newtons, roughly equivalent to
83 times the acceleration due to gravity.
Relative Uncertainty on Force = Relative Uncertainty of Δ velocity + Relative
uncertainty of Δ Time (since the mass was a given value, it was assumed to have no
uncertainty).
The absolute Uncertainty on velocity was designated to be ± 11.675 in/sec or ± 0.296
meters/second (in section 4.1). This must be multiplied by two since there were two
velocity readings involved hence:
Relative Uncertainty of Δ Velocity = 2(0.296)/ 13.25 = 0.0447
Since the absolute uncertainty on time was ± 0.01 sec in slow motion, it is equivalent to
0.00057 seconds in real time.
Relative Uncertainty of Δ Time = 2(0.00057)/(0.036) = 0.032
Total relative uncertainty on Force = ± 0.0764 or ± 7.6%
Graph 5: Slow Motion Head and Car’s motion together (distance)
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Graph 6: Slow Motion Head and Car’s motion together (velocity)
Graph 7: Slow Motion Head’s acceleration experienced at peak
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Please note that the peak acceleration point (34.979, 103.7) was taken from the
intersection of the extrapolation of the best-fit lines when the acceleration was changing.
The reason for this will be discussed in the Conclusion and Evaluation section.
4.4 Results in Scenario 2: With Seatbelt and Airbag
The starting time for this video sample was approximately 56.01 till 58.43 seconds (this
will be the range of the x-axis in Scenario 2 graphs). According to the distance-time
graph (graph 8) comparing the motion of the car and the dummy, the crash begins at
56.22 and ends at about 57.27 seconds. Since the dummy is strapped to the car, the
dummy’s head begins to decelerate at the same time as the car. Hence for the same
change of speed, the dummy’s head has a longer time period (this argument will be
further discussed in the Conclusion and Evaluation). The velocity of the dummy changes
from – 31.552 inch/sec to -1.929 inch/sec (graph 9) and the duration of this time period is
1.05 seconds in slow motion.
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Following the same reasoning as shown in section 4.3 for scenario 1, the calculated
measurements would be as follows:
Average Force = 992.5 Newtons
Average Acceleration = 220 m/s/s (22.5 times the acceleration due to gravity).
Peak Force = 3283.7 Newtons
Peak Acceleration (from the graph) = 729 m/s/s
Relative Uncertainty on Force = 6.4%
Graph 8: Slow Motion Distance travelled by the Car and Head with seat belt and air bag
during the collision.
Graph 9: Slow Motion Velocity of Head during the collision (with contact with airbag)
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Please note that the small jump in the point collection in graph 8 at point 56.6 seconds is
the cause of the unusual bump in the derivative graph. This value does not affect the final
calculation since an average of a larger number of points is taken.
Graph 10: Acceleration of Head at the time of impact
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(Peak acceleration would be the intercept of the two slopes drawn, point (70.07, 93.312
in/sec/sec in slow motion). In this graph, a different video sample was used to acquire the
acceleration (though it was the same scenario and the same car), and this would explain
the different time scale in the x-axis.
5. Conclusion and Evaluation
Analysis from the videos indicates that having a seatbelt on and an airbag during a car
crash reduces both the average and the peak force felt by the passenger during a collision.
In both scenarios, the original velocity was kept constant; hence, the change in
momentum for both cases was also constant as momentum is calculated from mass times
the change in velocity.
As visible from the distance-time graphs (5 and 8) the motion of the dummy’s head as a
function of time varies when the passenger is attached to the car with the use of a seat
belt. In graph 5, the dummy’s deceleration begins much later whereas in graph 8, it is
almost parallel to the deceleration of the car. In the non-seatbelt scenario, the dummy
continued to move at 30 mph even when the car had begun to slow down which later
made the dummy decelerate from the same velocity within a smaller period of time. The
effect of the seatbelt was to prevent the dummy from continuing at the same speed even
after the collision. Interestingly, graph 6 shows us that the dummy had continued on the
same speed (of 30 mph) even when the car was decelerating from 33 to 35 seconds, and
only experienced a change in speed at about 35 seconds and it’s deceleration lasted only
approximately half a second.
The effect of the airbag in the collision could arguably be different from that of the
seatbelt. On one hand, the seatbelt allowed the dummy to slow down with the car, on the
other hand, the air bag increased the stopping time even further when the momentum of
the dummy was changing from the original (mass*velocity) to zero. This can be seen
from the calculations in section 4, in the presence of an airbag and seatbelt, the force felt
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by the passenger was created by an acceleration 22 times the acceleration due to gravity
as opposed to the 83 times which is nearly four times as large.
A probable cause for an airbag could also be to increase the surface area, which the head
comes in contact with. Even 22 times the acceleration of gravity can be harmful if it is
only focused on a small area. As opposed to the force applied from the steering wheel, a
larger area of the airbag would reduce the pressure applied on the passenger’s head.
There is no concrete way of measuring the error for the values that were calculated
above, however, there are ways in which systematic and random errors could have come
into play. One possibility for a systematic error could be parallax that could arise if the
video camera is not placed directly perpendicular to the motion. It could change the
distance readings in the distance-time graph and further affect the remainder of the
graphs if the camera’s angle is changed during the video. Interlacing of video frames
could have been a source of systematic error. That is when the frame rate is doubled in
videos, which affects the perception of the motion. This might have an affect on the point
collection if it is not consistent throughout the movie. An additional systematic error was
seen to come from the Logger Pro software during the selection of points. Visible in
graphs 5 and 8, there was an unintentional jump in the distance-time point with its origin
from the software.
Random errors were probably prevalent in this investigation. Most significantly, the error
would lie in the point selection. It is possible for the points to not be consistently on one
point due to the visibility in the motion in the film. Another source of error could be
when the scale was set before the collection. In essence, there is a possibility of error
whenever a point was selected from the video.
With many possibilities of error, there are ways to improve for future experiments as
well. In order to reduce the possible, it would be advisable to carry out the raw data
collection multiple times so that an average from a larger sample cancels out random
error. It would be interesting to look at different types of car models to see if they provide
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the same force on the impact and this extension would give us a better idea on the
accuracy of the force values collected in section 4, perhaps different sized cars or the
length of the front of the car (the place where the car crumples) would have a different
impact on the passenger with or without the seatbelt or airbags.
6. Bibliography
Allain, Rhett. “Uncertainty and Video Analysis.” Wired (Nov. 2009): n. pag. Web. 1 Dec.
2011. <http://www.wired.com/wiredscience/2009/11/uncertainty-and-videoanalysis/>.
Brian, Joel. “Video Analysis Software and the Investigation of the Conservation of
Mechanical Energy.” Contemporary Issues in Technology and Science: n. pag.
Web. 8 May 2011. <http://www.citejournal.org/vol4/iss3/science/article1.cfm>.
Jones, Griff. Understanding Car Crashes. DVD.
“Road traffic injuries.” World Health Organization. N.p., n.d. Web. 8 Oct. 2011.
<http://www.who.int/violence_injury_prevention/road_traffic/en/>.
Young, Hugh D., and Roger A. Freedman. University Physics. Ed. Addison Wesley.
2007. N.p.: n.p., n.d. Print.
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