Projectile Motion: Medieval Edition

Catapults were originally created in the second half of the Third Age, first gaining huge international attention during the siege of Gondor by the allied troops of Mordor. Though the engineers of these catapults intended them to hurl large stones or other standard medieval weaponry, the troops of Mordor used the severed heads of prisoners to launch over the walls of the capital city, Gondor.

 Proof:

That was lovely, wasn't it?

As evidenced by the video above, the word catapult comes from the Greek words "kata" and "pultos" meaning "downward" and "shield", respectively. Katapultos came to mean "shield piercer" during the Middle Ages.

Materials: 

-2 6’ 2x4s

-2” and 2.5” wood screws

-8 mini-bungee cords

-2 6” bolts

-4 washers

-2 nuts for 6” bolts

-1” eye hooks

-spicy peanuts can

Procedure:

-2x4s were cut to desired lengths using circular saw.

-2x4 pieces were screwed together to form base and arm support structure.

-holes were drilled using half inch bit for axle and bump stop (bump stop requiring multiple holes for adjustability.

-arm was attached to structure.

-eye hooks were screwed into bottom of throwing arm and into sides of base.

-peanut can was attached to end of throwing arm.

- bungees attached to finish off the catapult.

*Special thanks to Robert for single-handedly building our group's catapult

**Even more thanks for building it to get disqualified 

If it hadn't been for a clause that our group neglected to adhere to during the creation of our catapult we could have been a serious contender for the launching competition in class.

Using the video analysis software from Vernier Software & Technology, we were able to graphically analyze and collect data from the launch of a tennis ball with our personal catapult. 

The original video looked like this:

Booooooring

But with a little time and magic you can make something cool like this:

And when you convert these in to some crafty graph-tys you get all this neat information:

Graph showing vertical position and velocity as a function of time. 

  Graph showing horizontal position and velocity as a function of time.

Range of projectile.

With this information we can calculate a few things:


What we know: 


Δx=5.194m
 Δy=.75m
Δt=.966s






Let's find the Launch Angle and Initial Velocity!


Δy=v_{y0t +}  ½at
.75=v_{y0(.966) +} ½(-9.8)(.966)
v_{y0=5.68 m/s}


Δx=vx0(t)
5.194=vx0(t)
Vx0=5.38 m/s


(5.38)² + (5.68)² = √61.1315 = 7.82 m/s
v0=7.82 m/s


tan-1=(5.68/5.38)=46.55°

Make it or Brake it (or if you're Abby just crash into it...)

OH SHOOT Y'ALL, YELLOW LIGHT!!!

You know the moment when you're driving down the road and suddenly the light changes from green to yellow?! You take that extra second to think about whether to slow down or continue through the light, before flooring it and running the red light anyway.

If you're an ambulance, yellow lights do not apply.

Wanting to take a more physics-ical approach to this issue, we stood on the corner of Balcones and 2222 gathering data (and eating fro-yo: thanks Berry Austin!). We timed the length of the yellow light and the length of the intersection (using state of the art measuring equipment, see video below) and used this to calculate how long it would take to get through the light safely.

DATA:

Length of the intersection: 58.25 feet

Length of yellow light: 3.8 seconds

Speed Limit (feet/sec): 51.3 ft/s

Using the equation: v=d/t we are able to determine the maximum distance away from the intersection that a car can be and still make it (safely) through the light.

*And by safely we mean that all parts of the car must clear the intersection by the time the light turns red.

Maximum distance away from the intersection when the light turns yellow: 136.72 ft

Time it takes to get through the intersection going the speed limit: 1.135 seconds

Time left over: 2.665 seconds

According to James Madison University, the average stopping distance of a car traveling 35 mph will take 135 feet to come to a complete stop, confirming that the yellow light at this intersection will give a driver enough time and room to stop (taking into account human reaction time) before the light turns red at the maximum distance. Note: when we set up this distance on Sunshine, Abby's car (RIP) was able to stop in about 2/3 of that.

BUT WAIT, THERE'S MORE!

What if you want to speed up to make it through the light? We came up with a function of position vs acceleration to determine if you're an "x" distance away, how much you have to acclerate to get through the light safely. Exciting!!!

Δx=v0(t)=.5at²

plug in the data and you get...

a=x/4.5

*Our data isn't perfect, but it's pretty damn close. We don't know how to calculate percent error but we know we're sorta wrong.

Bibliography:

"Signals, Signs, and Markers." Texas Drivers Handbook. Austin: DPS, 2008. Http://www.txdps.state.tx.us/DriverLicense/documents/DL-7.pdf. Texas DPS. Web. 25 Oct. 2011. <www.txdps.state.tx.us

"What to Do When the Light Turns Yellow?" Driver's Ed Guru. Web. 25 Oct. 2011. <http://www.driversedguru.com/driving-articles/drivers-ed-extras/what-to-do-when-the-light-turns-yellow/>.

"Ratio of Speed to Stopping Distance." James Madison University - Home. Web. 25 Oct. 2011. <http://www.jmu.edu/safetyplan/vehicle/generaldriver/stoppingdistance.shtml>.

Brought to you by contributions from viewers like you, Lonnie:

Straight pimpin' on the corner

Galileo's Lab: The Write (Wright?) Up

 In this lab, Galileo's "Inclined Plane" experiment was duplicated to prove that by setting up a simple lab with a ramp, a metal ball, and a stopwatch that the distance an object travels is proportional to the time squared. A relationship similar to Galielo's was proved in this experiment.

So, a pretty long time ago a guy you may have heard of before, Galileo, tested the laws of physics and acceleration, things that previous physicists and philosophers alike failed to consider during the Middle Ages and Renaissance, even Aristotle hadn't heavily considered acceleration in his theory on motion.  

Using materials that would, especially by today's standards, be considered elementary Galileo revolutionized the debate on motion and acceleration.  By using a simple board with a grove down the middle and a small metal ball, Galileo was able to "dilute" gravity and have the ball roll down the ramp slowly.

What Galileo realized is that the distance it takes an object to fall is proportional to time squared. In our lab assignment, we were asked to recreate Galileo's experiment to prove this law. Without further instruction, we set out in our groups and set up different ways to collect the most accurate data. The materials we used were:

• an aluminum channel
• a steel ball
• a stopwatch
• meter stick
• several books (to incline the channel and form a ramp)

Our group (Robert, Abby, and myself) created a set up by stacking textbooks on top of one another and recorded the time it took for the ball to roll down the ramp at 10 cm increments. 

PROCEDURE:

• Stack textbooks and/or reams of paper and place the aluminum channel on top to create a slightly inclined ramp, angle does not matter.
• Mark off 10 cm increments on the ramp, starting from the bottom of the ramp and working your way to the top of the channel.
• Place ball on the 10 cm mark
• Release the ball
• Record with a stopwatch how long it takes for the ball to hit the table (repeat this step about 3 times to get a precise and accurate measure of the time)
• Repeat until you've reached the last mark (for us it was 80 cm) and have accurately timed each drop

Our data came out to look like this:

Click on the image above to expand

What we found is when you square the time it takes for the ball to roll down the ramp you get a linear correlation to the distance the ball traveled, just like Galileo said we would! Well, not directly, but through his equation: 

Distance ∝ Time²

 

Click on the video below to watch a clip of how the marble roll works:

Look how artsy this is:

BIBLIOGRAPHY:

"Inclined Plane Experiment." The Galileo Project. Rice University, 12 Apr. 1995. Web. 14 Sept. 2011. http://galileo.rice.edu/lib/student_work/experiment95/inclined_plane.html.

Wright, Richard. Wright AP Physics: Galileo's Lab. 2011. Print.