grade 11 physics parachute lab report

7/23/2019 Grade 11 Physics Parachute Lab Report

http://slidepdf.com/reader/full/grade-11-physics-parachute-lab-report 1/16

Lab 3Design Your Own Parachute

May 20, 2015

SPH3UG-02 Period 4

For Mrs. Hall

Written by Eric Zhang



Design Your Own Parachute

Prelab

When an object is in free fall, it accelerates at a constant rate due to the force of gravity. The equation for

the force of gravity acting on an object is

F

g

= m g

whereF

g

is the force of gravity, m is the mass of the object, andg

is the gravitational field strength. The

gravitational field strength on Earth is approximately 9.8 m/s2 [down].

If the object is allowed to fall though a fluid, such as the Earth’s atmosphere, the drag force counteracts

the force of gravity. The equation for the drag force acting on an object is

F

= ½ ρ v2 C D A

whereF

is the drag force, ρ is the density of the fluid,v

is the velocity of the object relative to the fluid,

C D is the drag coefficient, and A is the reference area of the object. By definition, the direction of F , thedrag force is opposite to the direction of the object. ρ , the fluid density

is calculated by dividing the mass

of an amount of fluid by its volume. The fluid density of Earth’s atmosphere at SATP is approximately

1.2041 kg/m3. C D , the drag coefficient, is a value related to the object’s geometry, and is obtained

experimentally. A, the reference area is the area of a two-dimensional projection of the object.

As the force of gravity accelerates an object, its velocity increases, causing the drag force acting on the

object to increase as well. When the magnitude of the drag force is equal to the magnitude of the force

of gravity, the object no longer accelerates as the net force acting on the object is zero. This leaves the

object at a constant velocity called terminal velocity. The equation for the terminal velocity of an object

falling through a fluid is

V

t

= 2

where V

t

is terminal velocity, m is the mass of the object, g is the gravitational field strength, ρ is the

density of the fluid, A is the reference area of the object, and C D is the drag coefficient. This equation is

derived from the equation for the force of gravity and from the equation for drag force, using Newton’s

Second Law, which states that

F

net

= m a

where F

net

is the net force acting on an object, m is the mass of the object, and a is the object’s

acceleration. At terminal velocity, the net force is equal to zero, and is also equal to the vector sum of the

force of gravity and the drag force.

F

net

= m a

F

net

= 0

F

net

= Fg

+ F

F

g

+ F

= 0

F g – F D = 0

mg – ½ ρV t 2 C D A = 0

V t = 2

For the average person falling in Earth’s atmosphere, terminal velocity is approximately 56 m/s. However,

after a fall, it is not safe for a human to land on the ground at this speed. A safe landing speed for real



skydivers is less than 7-8 m/s, skydivers equipped with parachutes usually have terminal velocities that

are less than or equal to this threshold. When landing, the amount of force that the skydiver experiences

can be calculated using Newton’s Second Law.

F = ma

In this equation, m is the skydiver’s mass, while a is a combination of the skydiver’s terminal velocity withthe parachute, as well as the skydiver’s landing technique. This is because the acceleration experienced

by the skydiver changes the skydiver’s velocity from his terminal velocity to zero, in an amount of time

based on the skydiver’s landing technique.

a =0 –

∆

Skydivers with better landing technique may bend their knees more as they land, increasing the amount

of time used to accelerate the skydiver, thereby decreasing the acceleration and the force experienced.

The skydiver’s terminal velocity is also an important factor, as higher terminal velocities will result in a

greater acceleration when landing and therefore a greater landing force. In general, landing forces of less

than 7200 N are considered safe for skydivers, with greater landing forces being potentially fatal.

The purpose of a parachute is lower an object’s terminal velocity by greatly increasing the drag force

while only slightly increasing the force of gravity acting on an object, thereby ensuring a safe landing. A

parachute accomplishes this by having a large reference area and drag coefficient, while having a

relatively small mass.

Basic parachutes have a dome shape with a hole in the center. This is to increase the drag force and

stability of the parachute by preventing air from “bleeding” out of the parachute unpredictably.

Irregularly shaped parachutes often have air escape the concave shape by leaving from specific edges of

the parachute. This results in a force that pushes the parachute horizontally, in addition the force that

pushes the parachute upward. The circular dome shape helps prevent this with its regular and even

shape. The hole in the center of the parachute further prevents this problem by allowing air caught in the

parachute to escape without spilling out from the sides that could potentially be uneven.

The object being suspended by the parachute and the parachute are connected by cables called

“suspension lines”. For skydivers, these lines usually have a length equal to that of the diameter of the

parachute. The length of these suspension lines may affect the performance of a parachute in at least

two ways. Shorter lines can restrict a parachute’s ability to open completely, thereby resulting in a

smaller reference area and a weaker drag force. The length of the suspension lines may also affect how

much the object being carried by the parachute swings back and forth, and how straight the object falls

to the ground.

Purpose

The purpose of this lab was to determine how the terminal velocity of an object with a parachute would

be affected by the object and parachute’s mass and shape of the parachute, as well as how length of

string used to connect the parachute to the object affected the horizontal distance.



Procedure

First, the flat plastic garbage bags were cut into sheets of different shapes, each with a surface area of

approximately 0.098 m2. The different shapes included a circle, a square, a regular hexagon, and a regular

octagon. For the circle, four areas near the edge of the circle were covered with a square strip of masking

tape on both sides of the parachute. These areas were such that a line between the two opposite areas

was the diameter of the circle, and that these two lines were perpendicular to each other. For the square,

the areas covered by masking tape were the four vertices. For the hexagon, two opposite vertices were

covered, as were two areas on the sides of the hexagon, such that the line between these two areas was

perpendicular to the line connecting the two vertices. For the octagon, every second vertex was covered

with masking tape. A hole was then punched, using a hole-puncher, through the areas covered by

masking tape, along with the masking tape itself. These four sheets of plastic were the “parachutes”.

Finally, a circular hole of radius 1 cm was cut in the centers of each parachute, to improve the stability of

each parachute.

Next, a basket for the parachute was created using paper. It was in the shape of a hollow cube missing its

top face. The side length of the cube was approximately 5.40 cm and it was put together using masking

tape.

Then, four strings were cut and tied to the octagonal sheet of plastic through the four holes. The ends of

the strings were attached to the paper basket using masking tape. This was done such that the distance

between a vertex of the plastic sheet and the vertex of the basket connected by the same string were 30

cm apart when the string was straight.

Next, coins were placed into the basket such that the basket and parachute were measured to have a

mass of approximately 5 grams. The actual mass was recorded. The parachute was then dropped from a

height of 3.38m off the top of a stairwell, and its descent was recorded using an iPad camera. This

process was repeated with masses of approximately 10, 15, 20, 25, 30, and 35 grams.

The coins were then switched back such that the basket and parachute had a mass of approximately 10

grams.

The parachute was then replaced with the hexagonal parachute, and dropped off the top of the stairwell

again. The same camera was used to record the parachute’s descent. This process was repeated with t he

circular and square parachutes.

Next, the string was replaced with a string of length 35 cm, and dropped off the stairwell while being

recorded by the same camera. This process was repeated with strings of length 40, 45, and 50 cm as well.

The videos were then processed into usable data using Vernier Software and Technology’s Logger Pro

3.9’s Video Analysis. The points plotted using video analysis followed the bottom of the parachute basket.



Figure 1: How the Parachute Drop Was Recorded

Materials Used

plastic garbage bags

masking tape

cotton string

blank paper

hole puncher

scissors

scale

video recording

device

Observations

The points plotted directly from the analysis of the videos were placed onto position-time graphs usingLogger Pro 3.9. In the following graphs, the red X points show the horizontal position of the parachute,

while the blue Y points show the vertical position of the parachute.

Graph 1: Position vs. Time for Trial 1 of Terminal Velocity vs. Mass of a Parachuting Object, Terminal

Velocity vs. Shape of a Parachuting Object, and Horizontal Displacement vs. String Length



Graph 2: Position vs. Time for Trial 2 of Terminal Velocity vs. Mass of a Parachuting Object





Graph 8: Position vs. Time for Trial 2 of Terminal Velocity vs. Shape of Parachute





Graph 11: Position vs. Time for Trial 2 of Horizontal Displacement vs. String Length

*Graphs for Trials 3, 4, and 5 of Horizontal Displacement vs. String Length are not included as their data has been lost.

In order to determine the terminal velocity of the parachutes in each graph, the linear section of the Y

points of each graph was selected and a line of best fit was drawn. The slope of this line was the terminal

velocity of the parachute for the trial. Only the linear section was selected in order to avoid including the

period where the parachute was accelerating to terminal velocity, but was not at terminal velocity. In

order to determine the horizontal distances travelled by the parachutes in each graph, the length of the

curve drawn by connecting all of the X points of the graph was calculated. This was done by taking the

sum of the absolute differences between the first and second point, the second and third point, the third

and fourth point, etc. The horizontal distances travelled were equal to the lengths of these curves. These

values were compiled into tables of values and were later plotted onto graphs using Logger Pro 3.9.

During the trials, the camera shook a small amount, resulting in shaky videos. The angle at which thecamera recorded each trial was also not perpendicular to the vertical descent of the parachutes, due to

the camera being held near the floor. During some trials, the parachute also moved towards and away

from the camera in the lateral plane. This movement was not reflected during the video analysis, which

dealt only with the frontal plane.

Table 1: Terminal Velocity vs. Mass of a Parachuting Object

Trial Number Mass g) Terminal Velocity m/s [down])

1 5.6 1.245

2 10.0 1.089

3 15.2 1.419

4 19.7 2.0465 25.8 2.474

6 30.2 3.786

7 34.3 2.535

*The string length was 30.0 cm and the parachute was octagonal in each trial.



Table 2: Terminal Velocity vs. Shape of Parachute

Trial Number Parachute Shape Terminal Velocity m/s [down])

1 Octagon 1.245

2 Hexagon 1.249

3 Circle 1.017

4 Square 1.462

*The mass of the parachute and basket was 10.0 g and the string length was 30 cm in each trial.

Table 3: Horizontal Distance vs. String Length

Trial Number String Length m) Horizontal Distance m)

1 0.0300 2.003

2 0.0350 1.74

3 0.0400 2.437

4 0.0450 2.494

5 0.0500 3.108

*The mass of the parachute and basket was 10.0 g and the parachute was octagonal in each trial.

Analysis

Part 1: Terminal Velocity vs. Mass

From Table 1, it is possible to conclude that at least one of Trial 6 and Trial 7 is an outlier. This is because

of how different they are from the rest of the data. Trial 6 shows a terminal velocity that is much higher

than all of the other trials, and is approximately 50 percent higher than the second highest terminal

velocity of Trial 7. Trial 7 shows a very low terminal velocity that is almost the same as that from Trial 5,

despite the approximately 33 percent increase in mass.

Graph 1: Quadratic Relationship between Terminal Velocity and Mass with Trial 7 as an Outlier



Graph 1 shows the possible quadratic relationship between terminal velocity and mass if Trial 7 is marked

as an outlier due to how low its terminal velocity value was despite its much higher mass. According to

Logger Pro, the equation for this quadratic is

V t =5315 m/s

kg – 90.72 m/s

kg + 1.558 m/s

where V t is the object’s terminal velocity and m is the object’s mass. According to this equation, as the

mass of the object increases, its terminal velocity increases exponentially. However, because the vertex of

this equation’s parabola is at (0.008534 kg, 1.171 m/s), if the mass of the object is less than 0.008534 kg,

increasing the mass to an amount less than 0.008534 kg will actually result in a decrease in terminal

velocity. However, according to the equation for the force of gravity acting on an object

F

g

= m g

the force of gravity should increase as mass increases. Because the drag force acting on the object

remains the same even as mass increases, the increased mass results in an increased terminal velocity.

Therefore, it does not make sense that the quadratic equation used to model this relationship predicts

that the object’s terminal velocity could potentially decrease with an increase in mass. This means thatthis model must be incorrect. The relationship between mass and terminal velocity could however still be

quadratic, with errors causing trials involving less mass to result in higher terminal velocities. A quadratic

relationship would not however explain the relatively low terminal velocities of the fifth and seventh

trials, meaning that the relationship is likely not quadratic.

Graph 2: Linear Relationship between Terminal Velocity and Mass with Trial 6 as an Outlier



Graph 3: Square Root Relationship between Terminal Velocity and Mass with Trial 6 as an Outlier

If Trial 6 is marked as an outlier and is excluded from the trend, two models can be made. Graph 2 shows

a linear relationship being used to model relationship, while Graph 3 shows a square root relationship.

The linear relationship of Graph 2 does not explain why the terminal velocity of the seventh trial would be

so low relative to the fifth trial despite the difference in mass. The square root relationship of Graph 3

does however, as its equation is

V t = 13.9√

which predicts that as the mass increases, the rate at which terminal velocity will increase decreases. This

explains why the seventh trial’s increased mass could potentially result in a relatively low terminal

velocity. Therefore, it is most likely that the model presented in Graph 3 represents the relationship

between the object’s terminal velocity and the object’s mass most accurately.

Part 2: Terminal Velocity vs. Parachute Shape

In Table 2, the terminal velocities of objects with differently shaped parachutes are compared while

surface area and string length of the parachute are held constant. The circular parachute resulted in the

lowest terminal velocity of 1.017 m/s, while the square shaped parachute resulted in the highest terminal

velocity of 1.462 m/s. The hexagonal and octagonal parachutes resulted in similar terminal velocities of

1.249 m/s and 1.245 m/s respectively, with the octagonal parachute resulting in a slightly lower terminal

velocity in this test.



Part 3: Horizontal Distance vs. String Length

Graph 4: Linear Relationship between Horizontal Distance and String Length

Graph 4 shows the relationship between the horizontal distance travelled by the basket and the string

length of the parachute using a linear model. This linear relationship shows that the horizontal distance

travelled is directly proportionate to the length of the string. The equation for this line is

d = 0.05928ℓ - 0.01480 m

where d is the horizontal distance and ℓ is the length of each string.

Conclusion

In this lab, it was determined that the relationship between the terminal velocity of the basket with a

parachute and its mass can be expressed as

V t = 13.9√

whereV t is the object’s terminal velocity and is the mass of the basket and parachute.

It was also found that the circular parachute was the most effective parachute shape and resulted in the

lowest terminal velocity despite having the same surface area as the other shaped parachutes. The

square parachute was the least effective, and the hexagonal and octagonal parachutes were similarly

effective.

The relationship between the length of string connecting the parachute to the basket and the horizontal

distance travelled by the basket was found to be representable as

d = 0.05928ℓ - 0.01480 m

where d is the horizontal distance travelled by the basket and ℓ is the length of each of the four strings.



Discussion

Errors

In this lab, most of the measurements taken were imprecise. There were many sources of error that

contributed to this issue. One of the most significant sources of error affecting all trials was the lack of

camera stability. When the parachute drops were recorded, the camera was held, instead of being placed

on a camera stand or being propped up on the floor. This meant that the camera moved. These

movements of the camera were interpreted as movements of the object during video analysis, despite

the fact the object was not actually moving in such a way. Another significant source of error was the

camera angle. As seen in Figure 1, the parachute drop was filmed from the ground. This angle caused

vertical movements nearer to the ground to appear to be larger, while vertical movements farther from

the ground appeared to be smaller. During video analysis, this would have caused the object to appear to

be accelerating, despite the fact that it had already attained terminal velocity. To minimize these errors,

the camera should have been placed on a tripod or a similarly stable object, with its camera angle

perpendicular to the motion of the parachute, which in this case would have been perpendicular to the

ground.

Another issue was the few amount of trials that were done. Ideally, at least 10 trials would have been

done for each part of this lab, with several repetitions of each trial. The increased amount of data would

have helped nullify the effects of randomness and random errors. Because of the few amount of trials,

certain trials may not accurately reflect the true relationships between variables. A prominent example of

this was the sixth trial of Table 1. This trial reported an abnormally high terminal velocity. The reason for

this was likely just random chance, though it could also have been caused by faulty deployment of the

parachute. This trial could potentially have led to the conclusion that the relationship between terminal

velocity and mass was quadratic, as shown in Graph 1 of the analysis section, instead of the square root

relationship of Graph 3.

In Part 3 of the lab specifically, another error was the inability to record and report the object’s

movement in the lateral plane. The camera and video analysis techniques used could only report

movement in the frontal plane, meaning that any movement involving the need for depth perception to

record it would not have been reported. In Table 3, this resulted in skewed values for the distances

travelled by each parachute basket, as certain baskets may have travelled more distance in the horizontal

plane, but only distance travelled of the x-axis was recorded. To resolve this, another camera could have

been placed such that its camera angle was perpendicular to the lateral plane, which would have allowed

distance travelled of the z-axis to be recorded as well.

Part 1: Terminal Velocity vs. Mass

In the analysis section, three possible relationships were given based on the data from the lab. Thequadratic model was determined to be incorrect when the sixth trial was determined to be an outlier and

when the model predicted a potential decrease in terminal velocity as a result of an increase in mass. This

however left the linear model and the square root model. The square root model was only chosen over

the linear model because it explained the low terminal velocity of the seventh trial while the linear model

did not.



The true relationship between terminal velocity and mass is given by the terminal velocity equation

shown in the prelab.

V t = 2

If 2 is combined into a constant k , the equation becomes

V t = √

V t = 2√

This equation is compared to the equation obtained in Part 1 of the analysis.

V t = 13.9√

It is possible for k 2 to be equal to 13.9, depending on the value of2

, meaning that the square root

model of Part 1 of the analysis could be correct. To determine if the model is correct, one could

determine the value of 2, as it should be equal to k . However, this is value cannot be determined as it

is not known how the reference area of the parachute changes from its flat sheet form to its concave

form when the parachute is deployed. The drag coefficient is also unknown. Because these two values are

unknown, neither of the two values can be calculated, and the validity of the square root model cannot

be confirmed. The square root model can only be said to be plausible.

Part 2: Terminal Velocity vs. Parachute Shape

The results for Part 2 of the analysis show that the circular parachute resulted in the lowest terminal

velocity, while the square parachute resulted in the highest terminal velocity, despite all parachutes

having the same surface area. As was seen in the prelab, a factor contributing to this result was likely to

be “air spilling”. The evenly shaped circle may have prevented air caught in the parachute from escaping

through edges because there were no irregular areas along the edge of the parachute that allowed air to

escape more easily relative to the rest of the edge of the parachute. According to the equation for

terminal velocity, this decreased air spilling of the circular parachute would correspond with a higher drag

coefficient, while the square’s increased air spilling would correspond with a higher drag coefficient. The

octagon and hexagon would have drag coefficients between those of the square and the circle. This is a

plausible hypothesis.

Another factor that may have resulted in the circular parachute achieving the lowest terminal velocity

was its reference area when in use. The parachutes had the same surface area when flat, but the way

each parachute interacted with the string and the air made have resulted in the circular parachute

opening up the most, thereby having the largest reference area, while the square and the smallest

reference area due to difficulties in opening completely. A cause for this difference in the ability of the

parachutes to open completely could be the string lengths used in the lab. Each of the four strings used

during the shaped parachute test were 30 cm long. The circle, however had a diameter of approximately

36 cm while the square had a diagonal length of approximately 45 cm, both with the same surface area.

This meant that each 30 cm string would have had to angle outwards more in order to completely open

the square parachute, as the strings were attached to the corners of the square parachute. It is logical



that it would be more difficult to angle the strings more outwards for the square versus a smaller angle

for the circle. This would result in the circular parachute having a greater reference area, which would

explain why it resulted in the lowest terminal velocity. The square would have had the smallest reference

area. This hypothesis can also be applied to the hexagonal and octagonal, whose diameters were in

between those of the circle and square.

These results make the circle shaped parachute the most efficient based on surface area of material used

to construct the parachute and the length of string used to connect the basket to the parachute.

Part 3: Horizontal Distance vs. String Length

In Part 3 of the analysis, it was found that a greater string length resulted in a greater distance travelled.

This can be explained by considering the parachute’s string and basket system to be a vibrating

pendulum. A longer string length would have resulted in a greater amplitude of vibration as the

parachute was falling. This would have resulted in the parachute travelling a greater distance.

A linear model was made for this trend, though it may not have been correct. Because movement of the

z-axis could not be recorded, not all horizontal distance travelled was recorded and reported. This couldlikely have led to incorrect values of horizontal distance for each trial, meaning that the actual numerical

relationship between horizontal distance and string length would not be incorrect. This lab does however

show that there is likely to be a relationship between horizontal distance and string length, and another

lab, using solutions for errors found in this one, should be done in order to determine the actual

numerical relationship.

grade 11 physics parachute lab report

Documents