physics investigation - raahisha catapult is a ballistic device that stored energy, and on...

Physics Investigation

FINDING THE RELATIONSHIP BETWEEN THE LAUNCH VELOCITY AND THE ARM LENGTH OF A CATAPULT.

RAAHISH KALARIA

1 INTRODUCTION

A catapult is a ballistic device that stored energy, and on triggering, releases the energy by launching a

projectile. The different kinds of catapults like Trebuchet, Mangonel, and Ballista were used in ancient

and medieval siege engines1. A catapult is powered by any elastic material (e.g. Ballista), ropes (e.g.

Mangonel), or a counterweight (e.g. Trebuchet). In this investigation, a Mangonel has been constructed

and used.

Figure 1.1 – A simple Mangonel2

Mangonels are usually powered by the force cause by the twisting and coiling of a bundle of ropes,

called Torsion. In this investigation, instead of a Torsion powered Mangonel, a Mangonel which is

powered by elastic force, is used. The material used for powering is a simple rubber band.

The aim of the investigation is to see how changing the arm length of the Mangonel, affects the launch

velocity of the projectile.

The velocity of the projectile in its launch trajectory will be measured by first recording it at 24 FPS, and

then analysing the velocity using the motion analyser software, Tracker.

1 http://www.real-world-physics-problems.com/catapult-physics.html 2 http://webapps.yarmouth.k12.me.us/~mrice/2ndtri1011/nateg_tri2/Welcome.html

http://www.real-world-physics-problems.com/catapult-physics.htmlhttp://webapps.yarmouth.k12.me.us/~mrice/2ndtri1011/nateg_tri2/Welcome.html

2 RESEARCH QUESTION

Investigating the relationship between the arm length and the launch velocity of a projectile, using a

catapult, by systematically varying the arm length.

3 HYPOTHESIS

Increasing the arm length should cause an increase in the velocity initially, as the torque given by the

arm to the projectile will increase with the increase in arm length, which is nothing but the distance

from the fulcrum.

𝑇𝑜𝑟𝑞𝑢𝑒 = 𝐹𝑜𝑟𝑐𝑒 × 𝑃𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑓𝑟𝑜𝑚 𝑓𝑢𝑙𝑐𝑟𝑢𝑚

4 VARIABLES

4.1 INDEPENDENT VARIABLE Variable Unit How it was controlled

Length of the arm

Centimetres (cm)

The length of the arm was varied by extending the original arm by the required length of extension. The picture below demonstrates how the arm will extend.

Figure 4.1 – Arm design. The extensible arm can move along the fixed arm.

The length of the arm will be measured using a ruler, with the uncertainty of ±0.05 cm. The range of lengths take are from 15 cm to 23 cm. This is because if an arm length of lesser than 15 cm is kept, it would reduce the overall size of the Mangonel.

4.2 DEPENDENT VARIABLE Variable Unit How it was measured

Launch velocity of the projectile

Centimetres per second (cm s-1)

The launch velocity was measured using the software ‘Tracker’. In this software, the user is supposed to insert the video recording of the motion, and frame by frame, select the position of the object in motion. The lengths in the video will be obtained by reference. In this investigation, the reference was a metre rule with the uncertainty of ±0.05 cm. Thus reference is marked in the software as 100 cm of length. Thus, the total uncertainty in the analyser would remain to be ±0.05 cm, as in the software, we are selecting the whole ruler as a reference. The ruler was kept as parallel as possible to the trajectory of the projectile. When all the data, including the position of the object in all the frames of the motion is given, the software will give, upon selecting that option, the velocity-time graph of the object in motion, and also a table of time (t) and velocity at that time (v). The frame at which the ball was released from the arm was noted, and the velocity in that frame of time was considered to be the launch velocity.

4.3 CONTROLLED VARIABLES Variables Unit How it was controlled

Wind speed Kilometres per hour (km hr-1)

The experiment was conducted in a closed, contained, and air conditioned room with as less wind flow as possible, so as to reduce the resistance offered by air during the flight, or keep it constant all the time.

The Angle of launch (θ) Degrees (˚) The angle at which the ball is released from the arm was kept constant. If the angle is not kept constant, the velocity of launch would vary every time we change the length due to late or early release. There would then be two factors acting upon the launch velocity of the projectile – length of the arm and angle. The angle of launch does not directly affect the velocity, but if the time taken for the projectile to be launched is not same, there would be a variation in the velocity as a projectile launched earlier would not have gained the complete energy from the rubber band. To avoid this and keep the angle of launch constant, a restriction was placed in the movement of the arm so that at that position, the arm would move no further and the ball would be thrust ahead. This way, we can control the angle of launch.

Position of the Mangonel and arm

The position of the Mangonel was kept constant, and so was the position of the arm. The arm was fully extended to the maximum, and locked by an

automatic triggering system. This trigger-lock ensures that the arm remains in that fixed position for every launch, or else there could be deviation in the z-axis of the launch. If this happens, the exact position of the projectile will not be captured on camera, when it is placed perpendicular to the apparatus. The Mangonel was kept at a constant height also.

Rubber band The rubber band used was the same throughout the experiment. This is because every rubber band has a different Modulus of elasticity. Changing rubber bands could change the initial force acting on the arm, and thus the velocity.

Viewing angle (Camera angle)

Degrees (˚) The angle from which the motion was shot in the camera, was kept constant, along with the position of the camera. The camera was kept perpendicular to the setup, so that only one side of all the objects could be seen. This way, avoiding visual intrusion of the third dimension. The camera was mounted fixed on a tripod stand and was unmoved throughout the experiment.

Mass of the arm Grams (g) The mass of the arm was measured to be 18.8 grams. This was kept constant throughout the experiment as increasing or decreasing the mass would then change the velocity. This is the reason that an arm which could extend laterally was used, so that the mass would not very all the time the length was changed.

Mass of projectile Grams (g) The same projectile was used throughout the experiment. The projectile was a little spherical object with roughly 4.6 grams in weight. It was selected as it was neither too heavy, nor too light to be deflected from its trajectory. Changing the mass would also alter the launch velocity of the projectile.

Volume and surface area of projectile

Centimetre squared (cm2)

As the same projectile was used, the surface area which came in contact with air was also same. Thus, the air resistance would remain roughly constant in all observations.

5 APPARATUS

5.1 MANGONEL CONSTRUCTION MATERIALS Table 5.1 – Parts required for construction of catapult. Parts P9 and P10 are explained in detail further.

Part Code

Part Name Part Dimensions

(l x b x h) / (cm) ±0.05 cm Part

Quantity

P1 Plywood piece 20 x 2.5 x 1.3 2


P3 Plywood piece 15 x 5 x 1.3 1

P4 Plywood piece 2.5 x 2.5 x 1.3 2





P9 Plywood polygonal piece (Detailed specifications further in Section 6.1)

2

P10 Plywood axle support piece (Detailed specifications further in Section 6.1)

2


A1 Axle 6 cm long, 0.25 cm radius 1

A2 Axle 3 cm long, 0.25 cm radius 1

S1 Tetrix Servo Motor + Controller + Lego NXT 1 each

S2 Tetrix Flat Bracket 1

B1 Bushing 0.25 cm wide, 0.25 cm radius 2

B2 Bushing 0.5 cm wide, 0.25 cm radius 2

C1 Projectile holder 1

5.2 MANGONEL CONSTRUCTION TOOLS Table 5.2 – Tools required for construction of catapult.

Tool Quantity

Dremel with cutting tool 1

Power Drill with 5 mm drill bit 1

Sand paper/ Sanding tool 1

Strong adhesive for wood such as Fevicol 1

Paper tape 1

Ruler 1

Protractor 1

Rubber Bands 3

Hammer and nails 1

5.3 EQUIPMENT FOR EXPERIMENT Table 5.3 – Equipment required for conducting the investigation.

Equipment Quantity

Camera with minimum 24 FPS video recording 1

Tripod stand 1

Metre Rule 1

Laptop computer with Tracker software installed. (https://www.cabrillo.edu/~dbrown/tracker/)

1

Spherical marble (projectile) 1

https://www.cabrillo.edu/~dbrown/tracker/

6 PROCEDURE

6.1 MAKING THE MANGONEL The following process below shows how the Mangonel was designed. Reference to the parts code in

Table 5.1 is made. Please refer to the table simultaneously.

The figures of P9 and P10 are shown below in detail with their dimensions.

Figure 6.1.1 (a) (b) – Diagrams of P9 and P10 respectively

Steps:

1. Take a 1/2 inch thick plywood sheet and cut out all the Plywood pieces shown in Table 5.1 and Figure

6.1.1, using a Dremel. First, draw the parts using a pencil on the sheet and then proceed with the

cutting. Also, using the power drill and the 5 mm drill bit, drill a hole through the P10’s as shown in

Figure 6.1.1 (b).

2. Take both P1’s and P2’s and fix them using the furniture adhesive as shown in the images below. This

will form the basic frame of the catapult.

Figure 6.1.2 (a) (b) – P1’s and P2’s forming the basic frame of the catapult.

P2

P1

(a)

(b)

16.5 cm

11.7 cm

3.8 cm

3 cm 1.2 cm

1.2 cm

3.5 cm

3.5 cm

5.8 cm

3.5 cm

2.5 cm

2.5 cm 1 cm

(a) (b)

3. Next, after a few minutes of drying, attach P3 to the frame as shown in the image below.

Figure 6.1.3 (a) (b) – Step 3 of making the Mangonel.

(a) (b)

4. Next, after letting the model dry for a few minutes, take the P5’s and attach them to the model

firmly as shown. Secure all the previous bonds, along with this new one with paper tape.

Figure 6.1.4 (a) (b) – Step 4, attaching the vertical support structures.

(a) (b)

7.5 cm

P3

P5

5. After the P5’s are attached firmly, take P6 and attach it to the vertical structure as shown in the

images below.

Figure 6.1.5 (a) (b) – Step 5 of the procedure. Attaching the restriction (P6) to the vertical supports.

(a) (b)

6. Next, attach P10’s on P3 as shown in the figure below with adhesive, exactly 5 cm from each side of

P3. Secure the joints with paper tape and let the structure dry

Figure 6.1.6 (a) (b) – Step 6 of the procedure. Attaching the axle support structures (P10) to P3.

(a) (b)

P6

P10 5 cm 5 cm

7. Take the P9’s and glue the 3 cm side of the piece exactly below P3, on P5. Also attach the surface of

P9 in contact with the side of P1, with adhesive. Reinforce the joints with paper tape and let the

model dry.

Figure 6.1.7 (a) (b) – Step 7 of the procedure. Attaching the slant support structures (P9) from P5 to

P1.

(a) (b)

8. Next, take P7, and mark a point on it 6 mm from one side, and 8 cm on the other side, exactly on

the centre of the long face. Using the power drill and the 5 mm drill bit, drill a hole across the part

on the marked points, making sure it is straight. The finished piece should look like shown in the

figure below.

Figure 6.1.8 – Step 8 of the procedure – Drilling holes in the primary arm (P7) for axle attachment.

9. Take the axle A2 and insert it in Hole 2, so that it is perfectly centred. Fix the axle in the position by

applying glue over the edges which are in contact, and let it dry. The diagram is shown below.

Figure 6.1.9 – Step 9 of the procedure – Inserting the support axle (A2) in the primary arm (P7).

8 cm 0.6 cm

Hole 1 Hole 2

A2

10. Next, insert the axle A1 into one of the axle support structures (P10), and attach a bushing B2

through the axle on the part of the axle between the P10’s. Now insert the arm between the pieces,

and pass the axle from Hole 1 of the arm, and on the other side, attach the second B2 bushing. The

total number of parts between the P10’s now, on the axle should be both the B2’s, and P7. The arm

should be able to rotate about the axle with ease. Push the axle further so that it is now centred

between the P10’s. Once the axle is centred, fix its position be introducing two B1’s on both the

sides. The B1 can be fixed to the axle using the adhesive once it is in its position. The finished

product is shown in the figure below.

Figure 6.1.10 (a) (b) – Step 10 of the procedure. Attaching the primary arm to the Catapult.

(a) (b)

11. Take the P8, and using two rubber bands, fasten it exactly below P7. Pull P8 out of P7, until it there

is a gap of 2 cm between the edges of P7 and P8. On the part of the P8 which is coming out of P7,

stick C1, the projectile holder (cup shaped) and reinforce the joint with tape. This will require more

tape as it will have to withstand a lot of force. The completed structure should look like shown

below.

Figure 6.1.11 (a) (b) – Step 11 of the procedure. Attaching the secondary extensible arm (P8) to the

primary arm (P7) with the projectile holder (C1).

(a) (b)

B2

A1

B1

P7

C1

P8

P7

Rubber

Bands

12. Next, take the two P4’s, and attach them 1 mm from both the side of the arms, on P2. Apply tape

over the joint. The diagram of the completed structure is shown below.

Figure 6.1.12 (a) (b) – Step 12 of the procedure. Fixing the arm restrictions (P4’s) on P2.

(a) (b)

13. Stick the Tetrix servo on the Right side of the arm from the back, on P2, with the paper tape, such

that the spline of the servo is closer to the arm, and facing the projectile holder. Connect the Tetrix

Flat Bracket (S2) to the servo using the screws. Make sure that the minimum position of the servo is

lesser than the position of the servo when the flat bracket touches the arm. The finished structure

should look like the diagram shown below.

Figure 6.1.13 (a) (b) – Step 13 of the the procedure. Mounting the servo along with the Flat bracket

on P2, to create the trigger for the arm.

(a) (b)

14. Attach the YRB cable of the servo to the Servo controller in the Tetrix kit, and connect the Servo

controller to Sensor Port 1 of the Lego NXT. Power on the NXT, and connect it to a computer with

the RobotC software3 using the USB Cable.

15. Create a New program in the software, and copy the code given in the next page, into the input of

the compiler.

3 http://www.robotc.net/download/lego/

P4

P2

1 mm

separation

S2

Servo

C1

http://www.robotc.net/download/lego/

Code for the Servo Trigger:

//*Code Starts from here*//

#pragma config(Hubs, S1, HTServo, none, none, none)

#pragma config(Sensor, S1, , sensorI2CMuxController)

#pragma config(Servo, srvo_S1_C1_1, trigger, tServoStandard) #pragma config(Servo, srvo_S1_C1_2, servo2, tServoNone)

#pragma config(Servo, srvo_S1_C1_3, servo3, tServoNone)




task main()

{

while (true)

{

servoChangeRate(trigger) = 0; while ( nNxtButtonPressed == -1)

{

servoChangeRate(trigger) = 0;

servo(trigger) = 60;

}

while ( nNxtButtonPressed == 3)

{

servoChangeRate(trigger) = 0;

servo(trigger) = 255;

}

}

}

16. Once the program is copied, press the F5 key on your computer or press the “Robot - Compile and

Download” button in the compiler. On being prompted, save the program, and name it ‘Trigger’.

The program will be downloaded in a few seconds and you will be notified.

17. Disconnect the NXT from the computer and run the program in the NXT by navigating to

‘My Files’ ‘Software Files’ ‘Trigger’

18. Run the program. When you press the orange button, the Flat bracket should rise up, and when you

release it, it should press down on the arm, restricting it.

19. Next, connect the rubberband from A2, to the vertical support structure (P6), such that the

rubberband wraps over P6, and the other ends are attached to A2 from over the arm.

20. Next, lift the arm assemble upwards, while keeping a protracter on the side of A1, until the arm is

raised exactly 45° from its initial position, as shown in the figure below.

Figure 6.1.14 – Lifting the arm assembly to an angle of 45° for marking position of the restriction.

(Note: Rubberbands are not shown in the diagram)

21. Allign a ruler on the P10s such that its edge touches the arm at the position defined in the previous

step, and using a pencil, mark a reference line on the P10 using the edge of the ruler which is in

contact with the arm assembly. This is ilustrated in the figure below.

Figure 6.1.15 – Marking the reference line on P10s for placing the restriction on the arm.


45°

Arm Assembly

A1

Arm Assembly

Reference Lines

P10

Ruler

22. Next, take P11, and stick it with adhesive over the P10s using the marked line as a reference. After

the adhesive is dry, hammer a nail on each side of the piece so that the nail gois through P11, and

the P10s. The figure below shows the diagram of the step.

Figure 6.1.16 – Attaching the restriction (P11) for the arm on the P10s.


23. The catapult is now ready to be used for experimentation.

Nails

P11

Arm assembly

P10

6.2 SETTING UP THE EXPERIMENT The steps shown below highlight how the experiment is to be conducted, once the Catapult is ready.

1. Adjust the position of a table, such that it is against a surface which is white in colour (or any colour

easily differentiable from the colour of your projectile.

2. Place the catapult on one corner of the table. Make sure the sides of the catapult are perfectly

aligned with the table. Mark the position, and using double-sided tape, stick the Catapult in

position. The set up should look like shown in the diagram below.

Figure 6.2.1 – Attaching the Catapult to the table.

3. Next, take the metre rule, keep it parallel to the edge, and from the corner, mark a point at 1 meter.

Extend this point downwards so that the mark becomes a line which is clearly visible from a

distance. The distance from the line to the corner of the table, now represents 1 meter. This will be

used as a reference measurement in the Tracker software.

4. Mount a camera on a tripod stand, and adjust it at a distance of about 3 meters, such that the

camera lens axis is parallel to the table. Check the frame in the camera through the screen by

setting it into video mode, and make sure that the catapult is located in the bottom right corner of

the frame and the table forms the lower edge of the frame. Keep the area covered by the frame

quite large such that the table, and areas beyond the table are seen. Do not try to cover so much

area that the projectile is difficult to spot in the frame.

Figure 6.2.2 – Alignment of the camera

3 meters

Camera

Tripod Stand

Catapult

Table

Ground Level

5. Adjust the arm length of the catapult to the length desired (based on the trial number), using table

6.2.1.

6. Set the video camera to high speed shooting mode, turn on the NXT, and prepare the catapult.

7. Start the recording.

8. Place the projectile in the projectile holder of the Catapult, and press the Orange button on the

NXT, which will release the trigger.

9. Stop the recording.

10. Repeat this process five times with the same projectile and the same arm length.

11. From the camera, copy the video files to a PC with the Tracker software installed.

12. Launch Tracker. Once the software opens, the screen you will see is shown in the figure below.

Figure 6.2.3 – Main screen of Tracker

13. Next, go to File, and place the cursor on the ‘Import’ option, and when shown further options, click

on ‘Video’. You will now be prompted with a window asking you to select the video file. Navigate to,

and select the desired video file, and press ‘Open’. The software will now load the video file into the

interface.

14. Drag the slider shown below forward until the slider reaches the exact frame before the frame

when the ball is launched. One frame at a time can be skipped for positioning the slider at the exact

point, but pressing the buttons shown in the figure below. After the slider is at this position, right

click on the slider, and select the option which says ‘Set start frame to slider (###)’, where ‘###’ will

be the frame number. Now drag the slider ahead until the frame when the projectile leaves the area

covered by the frame. Right click on the slider again at this point, and select the option ‘Set end

frame to slider (###)’, where ‘###’ is again the frame number.

Figure 6.2.4 – View of the bottom bar which contains the slider for navigation in the video.

Slider Buttons to skip one

frame at a time

15. Next, lengths in the video have to be calibrated. Click on the calibration button shown in the figure

below, and from the drop down, place the cursor over the ‘New’ option, and select the option

‘Calibration Stick’, as shown in the figure below.

Figure 6.2.5 – Locating the ‘Calibration stick’

16. Completion of the previous step will result in the appearance of a blue line on the video, which is

the ‘Calibration Stick’. Drag the stick over the section which had been marked as 1 meter length.

Extend the Calibration Stick by clicking and dragging the crosses on either side, until they are exactly

on the two points which enclose the 1 meter distance. The video frame can be zoomed in by

scrolling on the display window. Once the stick has been adjusted, set the distance as 100, which

corresponds to 100 cm. Besides the box on the top where you can define the length, there is a box

labelled ‘Angle from x-axis’. Make sure this is 0°, so that the line is perfectly straight. The video is

now calibrated for length.

Figure 6.2.6 – (a) The Calibration stick (b) Boxes for changing properties of the Calibration stick

(a) (b)

17. Next, the projectile needs to be defined as a ‘Point Mass’, so that the software can track its position

automatically in the frames. Click on the ‘Create’ button in the toolbar on top, and select the option

‘Point Mass’. Now, zoom in to the frame so that the projectile is clearly visible, and press Shift and

while pressing it down, click on the centre of the projectile. Doing so, a red dot on the projectile will

be visible.

Figure 6.2.7 – (a) The ‘Create’ button (b) Defining a point mass

(a)

(b)

Crosses to change position

Define the length Angle of the Stick

Projectile

18. Now that the mass has been defined, the auto-tracking process can be started. Click on the

‘Autotracker’ button on the top toolbar. The Autotracker window will open. Now, press Ctrl + Shift,

and click at the centre of the projectile. Set the evolution rate to 50%. Press the ‘Search Next’

button on the Autotracker window. The software will search the next frame for the projectile. If the

action results into correct detection of the projectile, continue pressing the button. If the software

cannot find the projectile, simply locate it manually, and click on it while pressing Ctrl + Shift. Once

the process is complete, the complete trajectory of the projectile will be visible in different points

on the video frame. Note the frame number at which the projectile loses contact of the arm.

Figure 6.2.8 – (a) The Autotracker button (b) Locating a mass in Autotracker mode

(a)

(b)

Figure 6.2.9 – (a) Trajectory of the projectile (b) Frame when the projectile leaves the arm

(a) (b)

Projectile

19. Next, once the motion of the projectile in the frames is detected, the data has to be exported to any

spreadsheet software. For doing that, click on the y axis label of the graph that is visible on the right.

Select the option ‘Velocity magnitude’. The graph will be updated. Then, Right-click on the graph

area and select the option ‘Analyse…’.

Figure 6.2.10 – (a) Selecting the y axis label (b) Analyzing the graph for values

(a) (b)

20. On the new window that opened, select data given to the right of the window. Left click on the

data, and select the option ‘Copy Contents’. Paste the data in any spreadsheet software and save it.

If the last frame has no value of velocity for it, eliminate the reading altogether. In the spreadsheet,

also highlight the velocity value corresponding to the frame number which was noted in Step 18,

which was the time at which the projectile was released from the arm.

Figure 6.2.11 – Copying the data of the graph

Data Table

21. Repeat the procedure five times for each arm length. Change the arm length according to the table

given below.

Table 6.2.1 – Length of the arm of the catapult for each trial.

Trial No. Arm length (cm) ± 0.05

1 15

2 16

3 17

4 18

5 19

6 20

7 21

8 22

9 23

22. Record the observations/data from tracker in the format shown by the table below. Round off all

the values to 6 decimal points. Note: The table only accommodates readings for any one arm

length. In total, nine such tables have to be made for different arm lengths.

Table 6.2.2 – Format for recording the data from Tracker. Note: Highlight the row which contains

the velocity at launch.

Arm Length

(cm)

Frame

number Time (s)

Velocity (cm s-1)

Reading 1 Reading 2 Reading 3 Reading 4 Reading 5

1 0.000000

2 0.041666

3 0.083332

4 0.124998

5 0.166664

6 0.208330

7 0.249996

8 0.291662

9 0.333328

10 0.374994

11 0.416660

12 0.458326

13 0.499992

7 SAFETY GUIDELINES

Take extreme care while cutting out the wood pieces using a Dremel. Always wear eye

protection and heavy duty hand gloves. Wear a lab coat or a full sleeve t-shirt. Do the cutting

under adult supervision or have the pieces cut out from professionals or adults to desired

dimensions

Do not let the sand paper graze your skin.

Always polish and use the wooden pieces or else they can prick the skin easily.

Do not let the adhesive glue come in contact with your skin, or else it could cause irritation.

Do not aim the projectile on someone’s body.

8 DATA COLLECTION

8.1 QUALITATIVE DATA As the trigger is released, the rubber pulls the arm, and the arm collides with the restriction and is

stopped. The ball from the cup of the arm is released instantly and the projectile is shot in its trajectory.

The ball moves visibly slower after a certain arm length. Also, the range increases initially and then

starts decreasing after a certain arm length.

8.2 QUANTITATIVE DATA The collected data is summarized in the tables below. The highlighted velocities are the points in time

when the ball is launched from the arm.

Table 8.2.1 - Readings from Tracker for 15 cm arm length

Arm Length (cm) ± 0.05

Frame number

Time (s) Velocity (cm s-1)


15

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 60.539849 53.210328 57.615240 57.864187 50.311242

3 0.083332 281.562954 292.137735 239.861882 245.060943 224.113775

4 0.124998 562.285198 586.122482 576.636052 589.461599 579.177281

5 0.166664 625.500914 637.117738 657.893335 643.494744 648.751482

6 0.208330 624.613122 603.708088 629.318180 609.609239 614.518721

7 0.249996 594.783587 574.661100 596.889376 579.234673 583.468235

8 0.291662 564.502552 546.784583 569.928044 552.076598 556.263838

9 0.333328 537.421112 521.032766 544.830570 527.858349 533.472979

10 0.374994 514.421625 500.963153 520.888421 505.365278 510.227280

11 0.416660 493.067027 483.794685 501.810878 486.415902 489.402053

12 0.458326 473.523890 468.748565 485.649458 470.869425 475.770484

13 0.499992 460.067374 455.561687 471.105000 457.218619 463.416874


Arm Length (cm) ± 0.05

Frame number

Time (s) Velocity (cm s-1)


16

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 51.690853 63.690853 64.868490 69.194354 66.295843

3 0.083332 259.179111 297.746332 305.763733 289.428800 250.056670

4 0.124998 545.707503 545.723984 546.254676 547.184756 544.077704

5 0.166664 625.181943 603.816441 608.546060 607.461663 603.255251

6 0.208330 591.570200 575.462779 586.639567 565.403858 572.984325

7 0.249996 559.160235 558.781555 552.453021 547.664263 534.194489

8 0.291662 533.185703 522.754563 534.622457 537.462347 525.654338

9 0.333328 508.610365 507.316171 502.974213 498.444851 490.824405

10 0.374994 485.311691 483.463746 484.7852356 486.835553 487.386883

11 0.416660 466.914389 466.985059 465.338137 460.916974 461.450167

12 0.458326 450.712033 446.923459 449.732557 451.436678 452.844743

13 0.499992 437.412734 439.030929 435.7349526 436.036653 437.638296

14 0.541658 428.057590 425.543898 424.486631 430.973568 429.048673


Arm

Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


17

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 63.890624 62.173105 53.216195 68.502797 47.955668

3 0.083332 309.313213 302.165858 263.730642 276.037034 241.214960

4 0.124998 594.243010 589.292102 554.846993 534.930765 541.085486

5 0.166664 638.512175 640.191031 636.491330 637.071792 648.014366

6 0.208330 607.965201 607.757932 602.564425 606.223615 613.861968

7 0.249996 577.247457 575.899523 572.563313 576.927618 581.906230

8 0.291662 548.687613 547.116737 545.615950 547.689616 554.988383

9 0.333328 526.211036 524.461618 519.611649 523.691657 528.711005

10 0.374994 503.024373 501.473056 495.640141 501.952021 504.012975

11 0.416660 480.622309 478.307343 476.521248 482.279736 484.775715

12 0.458326 467.160038 463.583757 459.919236 467.820350 467.692920

13 0.499992 454.696176 450.039992 444.676414 454.843398 452.836664

14 0.541658 444.669958 439.876282 435.679005 446.4936763 443.460725


Arm

Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


18

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 54.164141 64.509354 55.674262 46.369290 63.554389

3 0.083332 261.557975 303.450019 262.375798 207.127088 324.914800

4 0.124998 563.745594 596.606389 567.778777 578.103044 604.324668

5 0.166664 655.448758 652.034833 637.407231 662.238081 667.244215

6 0.208330 622.150321 619.456353 634.314294 631.607080 602.930586

7 0.249996 589.264528 587.700991 603.792936 596.322568 570.333907

8 0.291662 559.580953 559.121865 576.133811 568.383744 542.363805

9 0.333328 534.176175 531.153694 554.249675 540.361889 515.714598

10 0.374994 508.575767 508.551759 529.958322 514.740512 489.606818

11 0.416660 487.884386 488.714908 505.977583 493.857101 469.021951

12 0.458326 471.306276 469.596190 492.302544 474.170988 449.790427

13 0.499992 452.258393 454.524422 479.189030 456.236450 433.834755

14 0.541658 440.580639 441.835643 436.2857743 444.114745 422.837253


Arm

Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


19

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 39.458343 36.138671 40.957221 32.751000 49.410381

3 0.083332 126.173026 150.278219 157.934547 119.681757 131.640981

4 0.124998 394.952638 451.422501 486.674461 413.020570 462.021085

5 0.166664 659.260079 647.546490 663.033663 645.173992 641.287866

6 0.208330 616.943929 614.100265 628.584005 641.645310 607.729640

7 0.249996 581.832716 582.694998 598.857677 611.581713 575.337547

8 0.291662 550.939993 552.177112 568.974190 581.403450 547.520193

9 0.333328 523.105968 522.107493 539.367590 552.407227 521.247721

10 0.374994 493.633850 497.695578 511.856441 528.044813 494.983879

11 0.416660 469.387783 475.869595 490.495647 505.565249 474.717334

12 0.458326 445.848089 456.686545 471.342236 482.959397 458.740671

13 0.499992 430.887204 436.551979 452.070344 466.794697 443.910883

14 0.541658 421.981812 419.468160 441.288236 452.961224 431.754689


Arm Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


20

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 41.937486 67.449790 51.929944 52.155084 42.955348

3 0.083332 145.543572 160.074045 131.927571 129.186983 150.478948

4 0.124998 424.419874 446.058658 413.732668 396.701513 434.016899

5 0.166664 632.107633 644.099048 629.147558 628.634152 639.915605

6 0.208330 615.262988 610.476019 621.457777 615.946312 621.328186

7 0.249996 585.716128 580.736845 602.470632 604.706909 590.273196

8 0.291662 558.017284 555.112147 572.888080 573.753328 562.038312

9 0.333328 532.067585 531.103268 547.650189 550.218080 535.625652

10 0.374994 508.075206 508.026320 522.466405 528.930351 514.317605

11 0.416660 485.768382 490.854481 500.163434 505.556803 492.670156

12 0.458326 470.874128 471.906648 483.103961 486.809532 473.149519

13 0.499992 456.191180 456.678806 466.636342 471.662978 459.846507

14 0.541658 441.327519 452.824139 451.718615 458.530894 450.909099


Arm Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


21

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 57.138135 58.013164 65.223989 62.077340 59.597547

3 0.083332 203.528329 200.156169 240.267057 210.422031 265.796711

4 0.124998 342.366193 355.842350 417.394006 369.913138 456.450627

5 0.166664 619.220470 621.460848 617.384347 606.112858 636.616612

6 0.208330 608.442615 606.549873 595.331341 609.687076 604.962091

7 0.249996 595.273730 585.639402 563.759521 579.825827 575.514697

8 0.291662 564.777772 555.316713 532.966310 550.767449 547.958124

9 0.333328 539.425373 529.619617 507.326889 525.870997 520.938980

10 0.374994 514.683272 505.461450 483.292687 502.968768 496.775388

11 0.416660 491.040303 482.153946 458.974010 481.435918 475.212185

12 0.458326 472.257228 464.305258 443.999909 462.497786 458.229091

13 0.499992 456.919254 449.194533 429.406036 442.019805 445.097973

14 0.541658 445.098069 436.549801 415.738404 431.354258 433.549967


Arm Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


22

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 53.312077 49.671956 59.424461 67.490583 45.015037

3 0.083332 259.355512 298.907107 220.562454 273.788498 148.748364

4 0.124998 533.029579 554.032361 492.701702 539.433594 418.218452

5 0.166664 617.038806 601.938541 605.198905 602.102120 608.252996

6 0.208330 584.362957 566.550051 573.159717 568.245139 587.021786

7 0.249996 552.325738 535.028410 541.314935 542.289924 557.118329

8 0.291662 525.791693 506.129879 511.289874 512.094202 527.637057

9 0.333328 498.835065 478.457960 484.645726 484.974438 499.787831

10 0.374994 475.333962 455.616348 462.915862 465.075176 474.886016

11 0.416660 454.440097 434.279648 439.557801 441.330113 450.923103

12 0.458326 434.428867 415.537535 419.626183 421.221960 431.586322

13 0.499992 420.818008 400.253396 405.252068 408.065945 416.331306

14 0.541658 411.661138 390.877805 389.187368 395.416433 403.054663


Arm Length

(cm) ± 0.05

Frame

number Time (s)

Velocity (cm s-1)


23

1 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

2 0.041666 53.029761 45.003978 43.912874 59.463396 36.440081

3 0.083332 149.376365 162.468875 153.615543 230.369131 112.842947

4 0.124998 497.318377 434.056429 415.352181 507.582381 362.688486

5 0.166664 606.056106 599.913292 580.682833 604.690450 584.972689

6 0.208330 581.600594 572.266731 554.192763 571.082664 587.069734

7 0.249996 548.630363 540.521590 522.300592 541.327710 556.800090

8 0.291662 521.757869 508.860908 492.168908 512.221683 526.945269

9 0.333328 495.726873 481.315515 464.425347 484.083575 498.703406

10 0.374994 470.417423 457.041752 440.362660 460.956316 473.053756

11 0.416660 450.462086 436.731575 418.102240 442.799257 452.186940

12 0.458326 434.868384 416.658482 396.245637 421.491268 436.441464

13 0.499992 421.990257 397.822955 382.733516 404.215105 424.739784

14 0.541658 410.440075 383.192825 376.416435 396.132576 421.227732

9 DATA PROCESSING

To process the data, the first step is calculating the uncertainty for the velocity reading. The uncertainty

of velocity depends on two other values – the uncertainty in displacement, and the uncertainty in time

measurement. This is because:

𝑉𝑒𝑙𝑜𝑐𝑖𝑡𝑦 =𝐷𝑖𝑠𝑝𝑙𝑎𝑐𝑒𝑚𝑒𝑛𝑡

𝑇𝑖𝑚𝑒

The software takes reference of the lengths using a meter rule which is placed in the experiment. Due

the software analysing the data, the length unit is considered to have no uncertainty. The time is

calculated by the number of frames per second, so the least count of time is simply the time difference

between any two frames, or the time of the first frame. The least count is therefore 0.041666 s.

Therefore, the uncertainty is 0.041666

2𝑠 = ±0.020833 𝑠.

Therefore, the uncertainty in the launch velocity can be calculated as follows:

Sample Calculation for Trial 1 Reading 1:

Time of launch = 0.124998 ± 0.020833 s

Launch Velocity = 562.285198 cm s-1

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =𝐴𝑏𝑙𝑠𝑜𝑙𝑢𝑡𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦

𝑉𝑎𝑙𝑢𝑒× 100

=0.020833

0.124998× 100 ≈ 16.67%

Therefore, the percentage uncertainty of the Velocity is also 16.67%

𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦 =𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑈𝑛𝑐𝑒𝑟𝑡𝑎𝑖𝑛𝑡𝑦

100× 𝑉𝑎𝑙𝑢𝑒

=0.020833

0.124998× 562.285198 = ± 93.714200 cm s−1

Therefore, the value of Launch velocity is 562.285198 ± 93.714200 cm s-1, at 45° angle.

The same step was repeated for all the readings of Launch velocity. The tables below show the

uncertainty in the Launch velocities in all the readings, at different arm lengths.

Table 9.1 – Launch velocities and their uncertainties for Arm length of 15 cm

Arm Length (cm) ±0.05

Reading number

Launch Velocity (cm s-1)

Uncertainty (± cm s-1)

15

1 562.285198 93.714200

2 586.122482 97.687080

3 576.636052 96.106009

4 589.461599 98.243600

5 579.177281 96.529547



Reading number



16

1 625.181943 78.147743

2 603.816441 75.477055

3 608.546060 76.068258

4 607.461663 75.932708

5 603.255251 75.406906



Reading number



17

1 638.512175 79.814022

2 640.191031 80.023879

3 636.491330 79.561416

4 637.071792 79.633974

5 648.014366 81.001796



Reading number



18

1 655.448758 81.931095

2 652.034833 81.504354

3 637.407231 79.675904

4 662.238081 82.779760

5 667.244215 83.405527



Reading number



19

1 659.260079 82.407510

2 647.546490 80.943311

3 663.033663 82.879208

4 645.173992 80.646749

5 641.287866 80.160983



Reading number



20

1 632.107633 79.013454

2 644.099048 80.512381

3 629.147558 78.643445

4 628.634152 78.579269

5 639.915605 79.989451



Reading number



21

1 619.220470 77.402559

2 621.460848 77.682606

3 617.384347 77.173043

4 606.112858 75.764107

5 636.616612 79.577077



Reading number



22

1 617.038806 77.129851

2 601.938541 75.242318

3 605.198905 75.649863

4 602.102120 75.262765

5 608.252996 76.031625



Reading number



23

1 606.056106 75.757013

2 599.913292 74.989162

3 580.682833 72.585354

4 604.690450 75.586306

5 584.972689 73.121586

Next, the average of the launch velocities for each arm length was calculated. One example is shown

below.

Sample calculation for Trial 1 – Arm length 15 cm: (Refer Table 9.1 for values)

=562.285198 + 586.122482 + 576.636052 + 589.461599 + 579.177281

5

= 578.7365223 ≈ 578.736522 𝑐𝑚 𝑠−1

To calculate the error in the averages, the Standard deviation of the values was calculated. The formula

for Standard Deviation is as follows:

𝜎 = √∑ (𝑥𝑖 − 𝑥𝑎𝑣𝑔)

2𝑛𝑖=1

𝑛

Therefore,

𝜎 = √∑ (𝑥𝑖 − 578.736522)

25𝑖=1

5

𝜎 = ± 10.545520 𝑐𝑚 𝑠−1

This process was repeated for all the reading sets. The results are displayed below.

Table 9.10 – Average Launch velocities for each arm length, along with the Standard Deviation in the

value

Arm length (cm) ± 0.05 cm

Average Velocity (cm s-1)

Standard Deviation (± cm s-1)

15 578.736522 10.545520

16 609.652272 8.974575

17 640.056139 4.673050

18 654.874624 11.409767

19 651.260418 9.392730

20 634.780799 6.889421

21 620.159027 10.936424

22 606.906274 6.228066

23 595.263074 11.677595

Thus, plotting the graph of Launch Velocity vs. Arm length, we get:

Figure 9.1 – Graph of Launch Velocity against Arm Length. Note: The horizontal error bars are too small

to be visible on the graph (± 0.05). The vertical error bars are the Standard Deviation of the average

value.

560.00

580.00

600.00

620.00

640.00

660.00

680.00

14 15 16 17 18 19 20 21 22 23 24

Lau

nch

Vel

oci

ty (

cm s

-1)

Arm Length (cm)

Launch Velocity vs. Arm length

Figure 9.2 – Graphs representing the Velocity vs Time plot for different arm lengths.

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6

Vel

oci

ty (

cm s

--1)

Time (s)

Velocity vs. Time (16 cm Arm Length)

0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.1 0.2 0.3 0.4 0.5 0.6

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8

Vel

oci

ty (

cm s

--1 )

Time (s)


0

100

200

300

400

500

600

700

0 0.2 0.4 0.6 0.8

Vel

oci

ty (

cm s

--1 )

Time (s)


From the data of the average velocities, we can also find out the length of the arm when the velocity

will be maximum, under the given conditions. If we look at Figure 9.0, we can see that the graph can be

divided into two parts – the increase and decrease. The graph can be plotted as shown below.

Figure 9.3 – Launch Velocity against Arm Length, showing the maximum velocity possible.

In the graph above, the point of intersection refers to the point at which the velocity will be maximum,

compared to the other arm lengths. This point can be found out by equating the equations of both the

lines.

∴ 26.182𝑥 + 189.08 = −13.987𝑥 + 915.4

⇒ 26.182𝑥 + 13.987𝑥 = 915.4 − 189.08

⇒ 40.169𝑥 = 726.32

⇒ 𝑥 = 18.08 𝑐𝑚

Therefore, the maximum velocity will be at an arm length of around 8.08 cm. The velocity at that arm

length can be found by substituting the value of x in any equation of the line.

∴ 𝑦 = 26.182(18.08) + 189.08

∴ 𝑦 = 662.49 𝑐𝑚 𝑠−1

Additionally, from the graph, we can also find out at what arm length the velocity will be zero, i.e., the

rubber band will not be able to pull the arm up. This can be found out by finding the x-intercept of the

decreasing slope. At the x-intercept, the y value is zero, which will give us the value of x at that point.

y = 26.182x + 189.08y = -13.987x + 915.4

560.00

580.00

600.00

620.00

640.00

660.00

680.00

14 15 16 17 18 19 20 21 22 23 24

Lau

nch

Vel

oci

ty (

cm s

-1)

Arm Length (cm)


Point of intersection

The graph of the x intercept of the decreasing slope is shown below.

Figure 9.4 – Graph showing the x-intercept of Launch Velocity against Arm length.

The x-intercept can be calculated by substituting the value of y with zero in the equation

y = −13.987x + 915.4

Therefore,

0 = −13.987x + 915.4

13.987x = 915.4

x =915.4

13.987

x = 65.45 𝑐𝑚

Thus, when the arm length is roughly 65.45 cm, the projectile will not launch

y = -13.987x + 915.4

0.00

100.00

200.00

300.00

400.00

500.00

600.00

700.00

800.00

14 24 34 44 54 64 74

Lau

nch

Vel

oci

ty (

cm s

-1)

Arm Length (cm)


x-intercept

10 CONCLUSION

Figure 9.1 shows the graph obtained when plotting Average velocity against Arm length. The graph

shows that the launch velocity increases as the arm length is increasing initially, up to 18 cm. Then, the

launch velocity starts decreasing gradually, at a lesser rate than it had increased initially. The hypothesis

was that the launch velocity would initially increase. It is partially proven correct. According to the

observations, the launch velocity initially increases, but then, after a certain arm length, it starts to

decrease as shown in figure 9.1. This is because the projectile will only launch if the following condition

is met.

𝐹𝑜𝑟𝑐𝑒 𝑟𝑒𝑞𝑢𝑖𝑟𝑒𝑑 < 𝐹𝑜𝑟𝑐𝑒 𝑔𝑒𝑛𝑒𝑟𝑎𝑡𝑒𝑑

The position of the rubber band on the arm of the Mangonel is fixed, and hence, the point at which the

force is applied is also the same across different arm length. Hence, when the arm length is increased,

the effort required to lift the arm and the mass of the projectile also increases. Therefore, after a certain

point, increasing the length of the arm further could start reducing the velocity gradually, as now the

force required to launch the ball will increase to a level which is closer to the force given by the rubber

band. Thus, the motion of the arm will be slowed down, and thus, the launch velocity should decrease.

We can also plot the graph to see how the velocity of the projectile changes in its trajectory with time,

at different arm lengths.

The graphs in Figure 9.2 show the change in velocity in the trajectory of the projectile. The general trend

in the graphs is a sharp increase in the velocity from 0, and then reaching a highest point, then

decreasing gradually. Initially, a certain time, there is rapid acceleration, because the projectile is thrown

from rest into motion immediately. Also, another point to be noted is that the launch velocity is not the

maximum velocity. There is an increase in velocity even after launching the projectile. After the increase,

the projectile gradually decelerates constantly up to some time, and then there is a very gradual

decrease. This can be due to the other forces which influence the projectile, like the air resistance,

gravity, etc. Then, the velocity of the projectile seems to stabilize and become constant until it is out of

the frame of reference.

Figure 9.4 extrapolates the graph further to the x axis, to estimate the length of the arm when the ball

will not launch altogether. This is because the mass of the arm would have increased, and the torque

requirement to lift the arm with the ball so far from the fulcrum, is not fulfilled by the current rubber

band.

From Figure 9.3, we can see that the third point (Arm length = 17) is out of the trend line. This could be

due to some random or systematic errors in the launch. The possible sources of these errors are

discussed in the next section.

11 EVALUATION

The table below shows some possible errors, their types, causes, effect on the investigation, and improvements to reduce the errors.

Error Type Source of Error Effect on the Investigation Improvements

Air resistance Random Air molecules in the atmosphere Changes in velocities by a certain random value in all observations

Taking more readings and measurements, and then averaging out the values.

Friction at the fulcrum

Systematic Contact of arm surface with the fulcrum placeholders, and contact of axle with the holes.

Unpredictable variation in the motion of the arm. Can reduce velocity to some extent if not kept roughly constant in all observations

Lubricating the surfaces with oil can reduce the friction effect. Also, using ball bearings in place of the holes for the axle can further reduce the friction.

Movement of projectile in z-axis

Systematic Due to minor dislocation of arm during launch, from its desired trajectory

This adds another direction to the motion, which is either towards or away from the camera. Due to this, in the frame that the camera is capturing, the exact motion is not recorded, but only its perspective from the other perpendicular axis is recorded. This can cause a marginal difference in the readings.

Fixing the arm firmly in its place while launch can fix the initial position. To avoid deviation in its trajectory, a smooth piece of some material can be placed along the movement path of the arm on both of its sides, so that the arm only has the space to move in the fixed space and direction. However, this would considerable increase the friction.

Error Type Source of Error Effect on the Investigation Improvements

Elasticity of the rubber band

Systematic After a certain amount of stretching, the rubber band can lose some of its original elasticity and the capacity to shrink back to its original length with the same force

Less force is applied on the arm of the catapult. This will reduce the velocity gradually after a certain number of observations.

-

Irregularity in the rubber band

Random The rubber band might not be evenly dense, and could have irregularities, leading to variations in elasticity at different points.

Inconsistent readings for the Launch velocity.

Buying a high quality rubber band, which is used for scientific purposes.

Uncertainty in the frequency of frames per second.

Systematic The difference of time between two individual frames

During the time between two frames, there is some motion happening. The more the frames available to analyse, the greater the precision of the velocity measurement. Due to little frame availability for one full motion, the velocity between two frames is automatically calculated by prediction. Also, at a higher FPS, we can determine the exact moment the projectile leaves the arm, and find a precise velocity due to greater number of previous samples of positions.

Usage of a higher FPS camera can be helpful. Also, providing enough lighting to the motion will help in decreasing the ISO requirement, thus making the image sharper and increasing the number of available frames.

Placement of the meter rule

Systematic The meter rule could be slightly diagonally placed and not exactly under the trajectory of the projectile

Wrong readings in Tracker. Placing the ruler after marking points on the table can reduce the possibility of it being inclined.

12 FURTHER INVESTIGATION

This experiment can be carried further by using the model to experiment with various other things like

the range of the projectile, the maximum height reached, the time of flight, etc., by varying factors like

wind speed, arm length, angle of launch, mass of projectile, height of the Mangonel, different rubber

bands, surface area of the projectile, etc. This model can open doors to various experiments and

investigations in the future.

13 BIBLIOGRAPHY

Physics of Catapults – History of Catapults

https://sites.google.com/site/physicsofcatapults/home/history-of-catapults

Physics of Catapults – How a Mangonel Works

https://sites.google.com/site/physicsofcatapults/home/how-a-catapult-works-the-physics/mangonel

Catapults.info

http://www.catapults.info/

Real World Physics Problems – Catapult Physics

http://www.real-world-physics-problems.com/catapult-physics.html

Yarmouth School Department – Catapult Physics

http://webapps.yarmouth.k12.me.us/~mrice/2ndtri1011/nateg_tri2/Welcome.html
https://sites.google.com/site/physicsofcatapults/home/history-of-catapultshttps://sites.google.com/site/physicsofcatapults/home/how-a-catapult-works-the-physics/mangonelhttp://www.catapults.info/http://www.real-world-physics-problems.com/catapult-physics.htmlhttp://webapps.yarmouth.k12.me.us/~mrice/2ndtri1011/nateg_tri2/Welcome.html

physics investigation - raahisha catapult is a ballistic device that stored energy, and on...

Documents