Meeting Log

MEETING SUMMARY 17-APRIL-2014, DURING SCHOOL HOURS
HARRY - HERE
DEREK - HERE

-Discussed materials to use
Counterweight Dumbbell weights
Base Square
Arms Tetrahedrically connected to base
Trigger Cam be observed later
-Discussed some physics behind a 25 foot throw
Water balloon is on average about 3.15 inches in

diameter and .227 kilograms
Kinematics are involved
The ideal 45 degree throw to get 25 feet is

26.4 meters per second
Centripetal/Circular motion involved
Confusing, due to there being two systems

that interact dynamically with each other
Development of simulation suggested
-Discussed the use of Calculus
can be advantagous to development

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MEETING SUMARRY 10-MAY-2014, 10:00 AM TO 4:00 PM
HARRY - HERE
DEREK - HERE

Because this was only our first out-of-school

meeting, both of us knew that we had to get cracking if we

wished to complete the trebuchet within our allocated time.

To make matters worse, we spent many of our hours trying to

figure out a bunch of equations we didn't need, such as how

the acceleration due to gravity accumulates to velocity as

the position of the arm changes. After a clarification of the

rules, we found out that the requirements for the physics

applications simply calls for an explanation of force and

energy, not applied equations. However, we did apply calculus

to derive kinematics into a dandy equation that gives how

velocious an object needs to be in order to reach a certain

distance, given that the projectile's initial direction is 45

degrees above the horizontal. Here's the derivation (Please

note: "V" is the initial velocity in both dimensions):
     ⌠t
δy = │ (V - 9.8u) du
     ⌡0

or, δy = Vt - 4.9t²

δx = Vt

Vt = 4.9t² + δy

V = 4.9t + δy/t

V = 4.9(d/V) + δy(V/d)

1 = (4.9d/V²) + δy/d

d = (4.9d²/V²) + δy

d - δy = (4.9d²/V²)

4.9d²/(d-δy) = V²
     ____________
V = √4.9d²/(d-δy)

So now we have V, which is both the X velocity and Y

velocity. The actual tangental velocity is that number times

the square root of two:
              ____________
Vtangental = √9.8d²/(d-δy)

Once we store 9 meters into the distance-to-fly d value and

-1.5 meters into the change-in-y δy value, we get the speed

our water balloon needs in order to fly a grade-winning

distance: 8.695 meters per second.

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MEETING SUMMARY 14-MAY-2014, 2:00 PM TO 4:30 PM
HARRY - HERE
DEREK - HERE

Today, the both of us actually knew how to work

efficiently on the content of the project; as such, we were

able to get more stuff done, such as completing the written

portions of the physics applications of Force and Energy;

that leaves just Projectile motion left.

In addition, we devised many possible solutions to

some building restrictions. Through the use of pencils and

playdough, the both of us could much more easily explain our

ideas to each other. Several great ideas arisen from this.

For example, building the frame of the trebuchet in the shape

of a pyramid (rather than a prism) allows for much more

structural stability. Also, we decided that the use of

implementing a piece of skateboard would most certainly help

with the question of what our swinging arm will use to

actually swing.

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MEETING SUMMARY 18-MAY-2014, 12:30 AM TO 8:30 PM
HARRY - HERE
DEREK - HERE

This day saw our longest meeting to date, ranging an

entire eight hours from start to finish. Derek's father,

Norman Goodwin, aided in certain phases of construction that,

theoretically, are too dangerous for a pair of inexperienced

11th graders to try, but for the most part, we accomplished

most of the non-powertool-related building. After a good read

of a few more catapult videos from Youtube, we tried to aim

the ratio of L1 to L2 at roughly 2 to 5.

After seven hours of long, hard, sweaty work, the

catapult was finally ready for fine tuning. Unfortunately, we

ran into two problems with our catapult in doing this. For

one thing, we did not have a half-pound test ball; we had to

use a full-pound ball comprised of sand to test our

contraption, leaving us to guess if our balloons can

withstand the G-forces of our sling. For another, our

catapult could only throw our projectile about six feet.

What's worse, the reasons for the failure to launch remained

unclear until we reviewed the footage of the launch. With a

camera set to shoot frames at 80 Hz, we estimated that the

ball at launch farted out at 5.3 meters per second, a far cry

from the 8.7 meters per second we need. At the very least,

our angle of launch wound up almost perfectly correct, at

about 44 degrees from the horizon.

There are two possible reasons why our mechanism

malfunctioned:

• The counterweight wasn't heavy enough. To fix this

problem, we'd have to add extra mass onto the counterweight

to provide maximum centripetal acceleration.

• The sling's ropes were too long. Because of the

shape of the counterweight, L1's angle stalls at about 60 to

70 degrees below horizon from .7s to 1s after launch. To

maximize centripetal acceleration increasing the speed rather

than changing the angle, we would want our projectile to

launch exactly when the arm starts stalling. Instead, our

projectile takes until 1s to launch, destroying some much-

needed inertia just to get to a 45 degree angle. Shortening

the lengths of the ropes would most certainly solve this

problem, since the entire sling would swing around faster.

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MEETING SUMMARY 21-MAY-2014, 3:50 PM TO 8:10 PM
HARRY - HERE
DEREK - HERE

Because this was our last day to finish anything, we

worked more vigorously during this meeting than any other

meeting. Harry and Derek completely finished the trebuchet,

with the most noticeable modification (other than the paint)

being the counterweight's gain in mass; now, the catapult's

counterweight appears disorganized, but is actually extremely

sturdy. Dubbed the "CataPortal", our trebuchet juts out its

projectiles a whopping 32 feet.

Testing and Modification

Testing and Modification

When we say and list these trial numbers, please note that while there are "three" trials on here, Harry and Derek actually tested their catapult dozens of times. These videos simply show samples of what happened.

Also, lucky for us, Derek offered to use his camera app to take high-speed video footage (8x slower speed) for the purpose of diagnosing whatever problem gets in our way.

Trial 1

Horay! It works! ...Partially.

Distance: 6 feet

One of the biggest problems we realized at first were the lack of restraints on the axle of the swinging arm. On our driveway, which sags westwards about 3 degrees, horribly threw the entire arm off center. In case this kind of angle also happens on the actual field, this should never happen. Not only does it compromise the structural integrity of the sides, it's dangerous---if that thing broke apart suddenly at launch, shards of PVC could scatter rampantly at unsuspecting victims.

Another thing that could have been off was the length of the rope. Notice how the counterweight's swing causes the arm to stall from 0.7 seconds to 1.0 seconds. It's as if the rope actually tries to force the projectile backwards during that time!

Our last possible reason we came up with was the lack of weights latched onto M1. After a few calculations involving the speed of the ball at launch, we've calculated the initial velocity to be just 5.3 meters per second, far short of the 10.3 meters per second we need. So, perhaps strapping random metal objects rigidly onto the counterweight will help? Let's find out.

Trial 2

There are quite a few modifications that our catapult underwent before the second trials:

• Cardboard/PVC sides were added onto the axle of the swinging arm to prevent excess swaying from side to side

• Our rope is only an inch shorter than it used to be---a series of unrecorded tests oddly shown that the rope length was just where it has to be

• A whole bunch of random metal objects were strapped to the counterweight via webbing and Harry's awesome knot tying skills.

<--We do what we must because we can.

So, did anything change?

What a disappointment. The release mechanism flopped.

Distance: None

The tests in Trial 2 were mind-boggling and confusing due to how inconsistent our catapult seemed to be. Sometimes our projectile would launch, sometimes it wouldn't. This video illustrates such inconsistency:

Distance: 19 feet, None

Harry and Derek argued on exactly what the cause of the misfires seemed to be in the first place. Derek suggested trying different angles for the projectile to launch from within the sling, while Harry wanted to duct tape over the hole near the top of L2. Either way, the debate raged for hours and the answer seemed indecipherable until a new video had to be taken just to work for the projectile motion sketches for this blog; it could have been done without it, but we tried it again for kicks. No one wrapped the sling up, nor was there any duct tape, so both thought the catapult would fail. BUT THEN...

Trial 3

By now, we painted our catapult orange and blue Portal colors.

WOOHOOOOOOOOO!

Distance: 28 feet

Neither one of us knew exactly what we did right at first, but we soon figured out that the secret was NOT spinning the sling in knots. Such a process prevents the releasing rope from flipping shut. As for duct tape, it just seemed unnecessary to add at that point. Don't fix something that works perfectly.

Excited, we ran run after run after run to make sure we're consistent this time.

Distance: 32 feet

Distance: 25 feet

Distance: 29 feet

At this point, Harry and Derek decided to stop construction; we had already done enough and were confident that our catapult would earn an excellent grade.

Process of Design and Construction of the CataPortal

Design and Construction

A picture of us on our first meeting day ↑

Samples of all the needless equations we thought we had to solve

The schematic for an idea to use skateboard pieces in the catapult. We dropped this eventually.

Pencils and Play-doh that had to be sacrificed. For science. And idea communication.

Cutting up a hunk of thick PVC that would serve as the base.

Right at the last minute, Derek's father allowed us to use a hunk of what was formerly a door as a smooth base for the sling to slide on during launch.

Our sling, in the making.

The arm uses two pipes near the axle to ensure sufficient strength, which is important since it's PVC. Yeah, we have a lot of PVC.

Our weights are supposed to hang onto the catapult by hinge; that way, M1 can swing M2 a bit faster. Later, we made a hole through the top parts of that metal.

At first, Derek doubted Harry when Harry stated that using PVC would work. As we both later found out, it worked wonderfully.

This modification to M1 effectively makes our catapult's counterweight swing.

After attaching M1, the swinging arm, and the triangular support beams.

The end of L2 had been painted orange to ensure safety.

Okee dokee then! Looks like we're ready for our first test! (Derek is holding the camera, as in a lot of these photos.)

Our catapult prototype.

Physics Applications: Projectile Motion

Projectile Motion

Unlike the previous two physics explanations, projectile motion deals with how far an object will travel through the air to reach its destination. There are numerous equations used to solve projectile motion problems, but the two equations which are most important in our case are the ones which involve the displacement of x and y.

δy = (Viy)t + .5at²

where δy is the displacement of the projectile's elevation,

Viy is the initial y velocity of the projectile's elevation,

t is time in seconds,

and a is the acceleration due to gravity

Basically, our first equation solves for how far the projectile will fall to the ground from the moment it's released. Using t, the function accumulates the projectile's velocity (hence the Viyt) and weight (hence the .5at²) onto the resulting displacement; one can thus think of it as an evaluated integral.

δx = (Vix)t

where δx is the projectile's horizontal displacement when it collides into the ground,

Vix is the initial x velocity of the projectile,

and t is the time in seconds

This next equation is far less complicated thanks to the fact that x velocities do not generally change for airborne objects. That way, all we need to remember is that velocity is how far something travels in a certain amount of time.

In a perfect world, the farthest launch angle for any projectile is usually about 45 degrees. Due to certain factors such as air resistance, that hardly ever happens except one is inside a vacuum, but because of both facts that there's not much of a difference between 43 and 45 anyways and that a 45 degree angle always means the absolute values of the x and y velocities are the same at start, we'll use a 45 degree angle as the angle we aim for.

After tossing those equations around together, we came up with an equation that calculates how fast a catapult must launch at a 45 degree angle in order to reach a certain distance, also given the vertical displacement of the projectile from launch to impact:

V = Sqrt( 9.8(δx)² / (δx-δy) )

where V is the magnitude of the 45 degree launch that must occur,

Sqrt() is a square root function,

δx is the horizontal displacement,

and δy is the vertical displacement

Now, let's plug in some values into this equation:

7.92 meters (26 feet) will substitute δx. Note that this includes both the required distance of 25 feet and the distance behind the front of the catapult which the ball launches from.

2.13 meters (7 feet) will substitute δy. Do not confuse this with the actual height of the catapult, as anything taller than five feet is unwanted.

One calculation later, we get the velocity needed to reach 25 feet: 10.3 meters per second. PLEASE NOTE, however, that we did not include air friction, so to make our catapult throw farther, we want to build the catapult so that it throws faster than that.

Physics Applications: Energy

Energy

The two forms of mechanical energy, Kinetic and Potential, come into play every time a catapult's trigger is pulled. Before we begin, let's quickly observe the difference between the two forms of energy:

KE = 0.5mv²

PE = mgh

while KE and PE are kinetic and potential energy,

m is the mass,

v is the velocity,

g is the gravitational constant,

and h is the height above ground

Kinetic energy is the energy of an object due to its movement, and potential energy is the energy of an object due to its position or arrangement of components.

Here's a diagram of our catapult at rest.

One might say that M1, or the counterweight, has a lot of potential energy just before launch because not only does it contain large quantities of mass, it is also fairly above ground. In contrast, the projectile's end of the arm has no potential energy, since it's on the ground. At the moment, we lack kinetic energy because nothing is moving.

Keep in mind, however, that probably the most important part of a catapult is its energy transfer. Take a look at the diagram below---of the same catapult, except just fractions of a second after being set off:

Now that things are moving, kinetic energy exists by feeding off of the potential energy in the counterweight. Even though the counterweight has a bit of kinetic energy itself due to its mass and slow movement, the projectile possesses a lot more because of the proportion of the length of L2 to that of L1, squared. Now that a significant amount of energy transferred from M1 to M2, the projectile can now jettison itself across whatever it's aiming at.

With that being said, a more massive counterweight and/or a longer L2 are both ideal for catapulting our projectile across large distances, for M1 could translate more potential energy to kinetic energy while the disproportionately high L2 further multiplies and squares the kinetic energy that fuels the projectile's travel.

Physics Applications: Force

Force

Some of the most important details to keep track of in the creation of the Cataportal are the forces being applied to its arm, balloon, and self.

First off, let's observe the equation for Force:

F = ma

where F is Force in newtons,

m is mass in kilograms,

and a is acceleration in meters per second squared

With this kind of logic, it's alright to call weight a force since Earth is something with mass that pulls other somethings with mass towards it with acceleration, resulting in the FORCE of gravity. Thus, the equation for weight is:

W=mg

where W is weight in newtons,

m is mass in kilograms again,

and g is gravity in meters per second squared.

On Earth, g varies from place to place from a variety of factors, like the contents of the crust or distance from the equator, but it can still be estimated as 9.81 meters per second squared no matter where someone is. That would make our eight counterweights (12.5 pounds) about 55.6 newtons.

For our 55.6 newton weight (labeled here as M1) to jettison the projectile (labeled as M2) at a sufficient speed, it must hold more mass than M1 times the projectile's arm length (L2) over the counterweight's arm length(L1), and then some.

To increase the range, one can either add more mass to M1, or increase the difference between L2 and L1.

After we reach the right speed, we need the right release at the right angle at the right time. Our catapult uses a slanted hook on the end of the L2 arm to force the catapult to release when a certain accumulation of inertia is reached. Once inertia and acceleration reach a value high enough, one of the ropes slips off from the metal rod within the swinging arm, thus releasing the ball.