The Catapult

A catapult was a medieval siege weapon used to throw large objects long distances.  We have built a small scale catapult to throw a tennis ball and have determined the physics involved.

Catapults were invented by the ancient Greeks, and were named by them too, as the word "catapult"is derived from the Greek verb "katapeltes" meaning "to toss" or "to hurl."  A catapult is defined by its ability to use torsion (twisting of an object) or tension (pulling or bending of an object) to throw objects. The first greek catapults used a giant crossbow to throw rocks at fortified structures or marching armies.  The crossbow version was then improved on and used by the Romans.  The Romans also created the stereotypical version of the catapult that we know today.  Called an "Onager" it had the rectangular structure, the spoon shaped arm, etc.  With the medieval period came the Mangonel, a larger more powerful  catapult with many variations.  It had continued widespread use until the invention of the trebuchet and finally became obsolete with the invention of gunpowder weapons.  No longer used as an instrument of war, catapults are now built for a variety of reasons.  You have probably heard of the "Pumpkin Chunkin'"contests.  But we have built a small scale catapult to determine the physics involved.

Our Catapult is about 14 inches by 14 inches and about a 12 inches tall and made of solid wooden blocks.  The arm is made of a broken lacrosse stick.  The power of the catapult comes from a 12 inch bungee cord that hooks to the end of the lacrosse stick, goes under the arm, and hooks to the back of one of the blocks.  This creates enough tension so that when we pull the arm back, the bungee stretches, and when we release the arm, the bungee snaps it forward.  There is a wire that is drilled into the sides of the blocks that stops the catapult's arm from going any further, but since the projectile has inertia it will continue to go forward and is thus in flight.

In order to understand the physics involved in the catapult we must first understand all the forces that act on the catapult and the projectile.  

Forces:

Normal Force that exists on every object.

Torsion or Tension: the force that is contained and will act on the catapults arm.

Throwing force: the force that is created when the torsion/tension is released and will act upon the projectile.

Gravity: once the object is thrown, gravity is the only force that will act upon it (besides air resistance, which we will ignore).

Now that we understand the forces involved, we can find out their relations.  The components of this relation are as follows:

1. The angle at which the catapult throws the projectile relative to the horizontal that is the ground. Represented by theta: 


θ

2. The velocity that the object has the instant it leaves the catapults arm. 
VO
 This is a vector that can be divided into vertical and horizontal components.

VYO ,VXO

3. acceleration due to gravity: 9.8 m/s/s.




Time, at any point in the projectile's arc is represented by 


T
The vertical and horizontal displacement of the projectile at any time in its arc is represented by 


ΔY, ΔX

We want to find the horizontal and vertical displacement of the projectile.

This is found by finding horizontal and vertical vectors of the original velocity vector.


These can be found by the following formulas:

Vxo = Cos(θ)*VO

VYo = Sin(θ)*VO

The horizontal component is the easiest, the horizontal velocity will always be the same for a projectile, so we multiply the horizontal velocity by the time the projectile is in the air.  Represented below:

ΔX=Vxo * T

The vertical component is a little harder, since the vertical velocity is always changing due to acceleration due to gravity.

ΔY=Yyo * T + .5(-9.8)*T^2

these formulas can be combined to create two master formulas:

ΔX=Cos(θ)*VO* T

and

ΔY=Sin(θ)*VO * T + .5(-9.8)*T^2

These two formulas will be able to derive any component of projectile motion.

We now understand the history behind a catapult.  We understand the forces that affect it.  We understand the the variables that represent and measure these forces, and the formulas that express the relationship between the forces. The only thing left to do is actually launch the thing!  Will get back to you on how that goes.

Gas or Break!?

How do you time that annoying yellow light right without breaking any laws?  Well, right now we are going to find out!

Well, lets take your average badly timed intersection:

This is the intersection of Burnet and North Loop.  I don't go that way much but Jack Houtz says it's bad so I'll take his word for it.  

Just to give you a reference, going West to East the intersection was measured to be 25 meters (82 ft)

The intersection's speed limit is 30 miles per hour.  Converting this to meters per second will make this problem a lot easier.

30mph * 5280ft / 3600 sec / 3.28 ft  = 13.4 meters per second

We also know that the yellow light at this intersection is a duration of 4 seconds.

Lets say a vehicle approaching the intersection at Burnet and North Loop.  That vehicle is going the speed limit, 30 mph or 13.4 mps, and will continue to go the speed limit throughout the event.  

This vehicle, at 13.4 mps , will travel 53.6 meters in 4 seconds.

The City of Austin Connection states that a red light violation will occur when a vehicle enters an intersection after a light changes from yellow to red.  So the vehicle can be a maximum of 53.6 meters at the exact time the light turns from green to yellow to legally make the light.

So if a vehicle, going the speed limit, can legally make the light if it crosses the point at 53.6 meters at the same time or before the light changes from yellow to red.


Now lets say that vehicle driven by someone that is not in a hurry to get somewhere, and they decide to stop.


We found that a standard vehicle traveling at 13.4 mps is able to stop safely and effectively in a minimum of 11.5 meters.


Thus a vehicle traveling at 13.4 mps and approaching the intersection would have to begin stopping 11.5 meters away from the front of the intersection in order to break safely.

With these two points, the farthest point a vehicle needs to reach to make the light and the closest point a vehicle can reach to stop effectively.  We can see an area between these two points is 42.1 meters.

This area represents the distance a vehicle driver has to decide whether to legally make the light or safely stop.  At the speed limit, this distance takes 3.1 seconds to traverse. 

 Say we give the vehicle driver a 1 second reaction time.  This means that the driver takes 1 second between perceiving the change in light color, and beginning to decide what to do.  Thus the driver has 2.1 of that 3.1 seconds to decide whether to go or stop.

Thus we have discovered the maximum distance from the intersection a vehicle can be to make this red light, the minimum distance the vehicle can be to comfortably stop, and the time a driver actually has to make a final decision.


Sources:


"City of Austin - Red Light Cameras - FAQ." City of Austin - Austin City Connection: Home Page.    Web. 25 Oct. 2011. <http://www.ci.austin.tx.us/redlights/faq.htm>.


Calculators, Converters, Apps and Source Code. Web. 25 Oct. 2011.        <http://www.csgnetwork.com/stopdistinfo.html>.

Proving Galileo's Time-Squared Law

 The Italian Physicist Galileo proved that the distance an object falls in a medium is directly proportional to the time it takes the object to fall that distance squared.  I created an experiment with a ball and a ramp that proves this law.

8/8/11

McCallum High School AP Physics 3^rd Period

Galileo Galilei, born in 1564 and died in 1642, was an Italian astronomer, philosopher, physicist, and mathematician.  He was responsible for many breakthroughs in physics, mathematics, and astronomy that were central in the Scientific Revolution and foundational in science today.  He is considered “perhaps more than any other single person, responsible for the birth of modern science.”  Last week I devised an experiment that proves Galileo’s theory that the distance an object falls in a medium is directly proportional to the time it takes the object to fall that distance squared.  The experiment used a metal ball and a v-shaped ramp to replicate the object and distance fallen.

Materials:

1 V-Shaped Ramp

1 Metal Ball

1 Book or other flat object to incline the ramp

1 Meter-stick with centimeter increments

1 Stopwatch

Procedure:

1.    I placed the meter stick within the ramp and marked with a pencil every .1 m (10 cm) from one end of the ramp. 

2.    I situated the marked ramp on a significant point on top of the book (or other flat object) so that the angle of the ramp was exactly the same throughout the experiment.

3.    I placed the metal ball starting at the .1 m mark on the inclined ramp.  I let go at the precise time I started the stopwatch.  Then I stopped the watch when the ball hit the end of the ramp.  This gave me the time in seconds that the ball took to fall .1 m.  I recorded this time in a table.

4.    I repeated this procedure 5 – 10 times and averaged the results to get the mean time in seconds it took the ball to fall .1 m.

5.    I repeated these two steps for the rest of the increments (.2 m – .8 m)

6.    When I had gathered data for all the increments ( .1 m – .8 m).  I put them into the table. 

7.    I squared the average time in seconds for all the increments to get the average time in seconds squared (s^2).

8.    I put the time in seconds squared and the distance fallen in a table.  Then I used those numbers to create a graph. The distance fallen was the independent (x) variable while the time in seconds squared was the dependent (y) variable.

This gave me the proportion of Falling Distance to Time Squared.

The proportion is represented in this graph:

By making the increments proportional it allowed me to easily see if there was any correlation.  By repeating the timing process I was able to average many numbers to get a value that is closer to the true value.  This resulted in little experimental error, as seen in the graph above.  When I reviewed the graph, I saw the proportion had a positive and relatively direct correlation.  Given that experimental error is probable, the proximity to direct correlation was too close to regard as coincidence.  So I concluded that distance is directly proportional to time squared.