Trajectory

Definition

The path a projectile travels in motion.



Yes, that is really a person being shot out of a cannon.

A Few Comments Before We Start
There is going to be a lot of math in this section. Do not freak out. The math will not hurt you. Most of this is very much common sense oriented. Don’t think of it as math; think about how you threw things last week with Levers. The numbers just stand behind what you already know in your head.

CAN HAVE FUN WITH PROJECTILES!


How it works

As the projectile moves its velocities push it both up and across. It will always have both velocities for this reason.

When the projectile goes up, it fights the force of gravity until its vertical velocity is zero and it stops going up.

The vertical velocity is zero only for a second when it reaches the maximum height. Then the projectile is falling and accelerating with gravity.

The entire time the vertical velocity of the projectile is changing, the horizontal velocity is staying the same. The projectile is always moving horizontally. For this reason it always has the same horizontal velocity. At every moment the projectile is moving, both velocities are always acting upon it. Whether it goes up and across or down and across it is doing two things at once, meaning two velocities. Even when one velocity is zero it is acting upon it, because the projectile isn’t moving!

The Velocities

Since the horizontal and vertical velocities control the path of the projectile they are what you need to figure out for trajectory. If we know how the projectile is being pushed, we know where it is going to go.


The velocities of a projectile are a bit trickier. Since the projectile is symmetrical, we only have to deal with half of it to find both velocities. The horizontal velocity is always the same and the vertical velocity is effected by gravity (which we know) after the half way point.

Getting Crazy!


THE GREAT SOH CAH TOA

Write this down! It is UBER important!

.....

No really. Write it down.

Copy this on a piece of paper.

Soh- sin = opposite over hypotenuse.

Cah- Cos = Adjacent over total velocity

Toa- Tan = opposite over adjacent.

Your opposite and adjacent sides, as well as your hypotenuse could also be called vertical velocity, horizontal velocity, and total velocity.

For more information on this assingment, see Mr. Evans.


How much time?

To figure out how much time the projectile will take to reach its maximum height use this formula:

(Final velocity - vertical velocity)/acceleration


Don’t freak out. This is NOT as hardcore as it looks. You already know all the numbers.

Acceleration means a CHANGE IN velocity. When the projectile is going up, it is slowing down. It is changing its velocity due to gravity pulling on it. It is slowing down at the acceleration of gravity.

You know the acceleration of gravity = 9.8 meters/per second2
Remember, you are slowing down so the value of acceleration is NEGATIVE.

To figure out how high the projectile will go to reach its maximum height use this formula:

(Vertical velocity x time) + (.5 x acceleration x time2 )

Again. Don’t freak out. This is NOT as hardcore as it looks. You already know all the numbers.

Let’s Just Think About It….

8.34 - 4.17 = 4.17

You know the vertical velocity. You already figured it out with SOH, CAH, TOA. Remember, we fired the projectile at 65 degrees. Just fill the number in.


Time refers to the specific moment in air you want to find the height of. You just figured that out. At what time is the projectile at its maximum height?

.92 seconds

How Far Will It Go?

To figure out the distance the projectile will travel use the formula:

Distance = Horizontal velocity x time

By now, this one should look easy.


Let’s Just Think About It…

Distance = 7.76 meters
For time, you need to figure out what you want to know.
You already know the horizontal velocity. You figured it out earlier with SOH, CAH, TOA. The projectile was fired at a 65 degree angle so the horizontal velocity is 4.22 meters per second.


How much distance does the projectile travel before it hits the ground?
The trajectory is symmetrical. If it reaches its maximum height half way through the total flight, multiply the max. height time by 2 and you get the total flight time.
92 x 2 = 1.84

The “Evolution” of the Catapult

The Ballista

First invented in 400 B.C. in Greece
Dionysus of Syracuse
Used torsion tension with 2 flexible LEVERS!
At first, not very powerful.
Launched darts





The Greek and Roman Battlefield


The Good & Bad

The Pros
Allowed targeting of individual soldiers – accurate
Universal Joint – fast trajectory changes
VERY advanced for time
Used both for support and siege tactics.
Mobile

The Cons
Complicated and required maintenance
Not very powerful at first.
Compromise between bolt weight and velocity
Required technology and resources

Random pictures!






The Gastraphetes

The original, primitive crossbow
Handheld Ballista
Called the “Belly Bow”





The End of the Ballista
Rome conquers Greece…make improvements and use the Ballista throughout kingdom
Rome falls…and the resources and skills required to build the Ballista goes with it.

Bye, Bye Ballista!

The Magonel

Background of Mangonel

THE CATAPULT – modern image
Alexander the Great – 400 B.C.
More basic than the ballista
Used torsion tension from rope BUT eliminated the flexible levers
“Savage” compared to ballista

The Good & Bad
The Pros
Cheap & easy to build
Powerful
Able to throw heavy, large objects far distances.
Excellent as a siege weapon against fortifications
Ammunition was available






The Cons

Low accuracy
Cumbersome
Unwieldy and hard to manage on the battlefield
You had better understand trajectory!


The Ammunition


And other stuff like this.

Up next:
The Trebuchet

Simple Machines: Levers

A lever is a simple machine consisting of a rigid bar pivoted on a fixed point and used to transmit force, as in raising or moving a weight at one end by pushing down on the other. Force is the amount of effort being applied either by your hand, body weight, or the weight of another object. Force is what is pushing or pulling.

First Class Levers:




Think of two other examples

Second Class Levers:



Or even this



Identify where the Fulcrum, Applied Force, and Load are in this image. (Hint: Do a push up or two to figure it out.)

Third Class Levers:



Explain how a shovel is a third class lever. Be sure to use the words Fulcrum, Force, and Load.

The Law of the Lever

Magnitudes are in equilibrium at distances reciprocally proportional to their weights.(Science) Forces are balanced when the heavier object is closer to the fulcrum and the lighter object is farther away. (English)

The Law of the Lever Explained
Be Calm Kids, The Scary Math Won’t Hurt You!

Force of F1 X Distance from the Fulcrum = Force of F2 X Distance from the Fulcrum


They are balanced because they are equal.


Mechanical Advantage

The mechanical advantage is how much the machine multiplies your force. Basically, how much the machine has helped ease the work.

Mechanical Advantage for Levers = Length of the effort arm / Length of the resistance arm.


To move an object… Distance to be raised x mechanical advantage/distance to be pushed.


Putting It All Together

While driving ever so cautiously home from helping blind, orphaned children learn to read your car is hit by an asteroid. Amazingly, the impact only smashed the trunk causing your car to be pinned in place but otherwise in perfect working condition!You are stuck and must get home to diligently work on your homework for Mr. Evans’ physics class. Just then you notice a large, astonishingly indestructible, board from a local construction site. You place the board under the boulder creating… Some quick estimation and you decide the boulder weighs 600 pounds. Some quick charisma and you convince the extremely strong bodybuilder walking by to help you.

How far are you going to push the board under the boulder to be able to use the lever?!?

How far is the bodybuilder going to have to push down for you to drive off?!?


Will you be able to get home to finish your physics homework?!?

Hint: The bodybuilder can exert up to 150 lbs of force, the board is 10 feet long and you have to lift the boulder 1 foot off the ground to get your car out.
There are multiple answers as long as you can explain them.

Answers
The boulder needs to be raised 1 foot off the ground.
Yay! Your homework does get done!

The moral of the story is…
Even under extreme circumstances your homework should always get done.

Your mission, should you choose to accept it, is to kill this.



Thats right, this here's a Wooly Mammoth. You all know what those are right? Well, its kind of like an elephant bred with a dinosaur. Its huge. What do you think you're going to kill it with? A gun? Alright, problem solved. But theres one fatal flaw with that plan... You live 13,500 – 9,000 years ago. And we all know they didn't have guns back then. But that didn't stop them from taking this goliath down, you see, there was a group of people that used to take these things down without shooting them. They were Clovis People.


Things to know about Clovis People:
-Bering Strait Land Bridge
-13,500 B.C.
-Paleo-Indian Hunters – favored hunting large game on the edge of the grasslands.
-First known group to populate the Americas?


Clovis Technology
-Clovis spear point – this item literally defines their society.
-Fluted rock spear point.
-Mystery of Origin & downfall and demise.
-And one more VERY IMPORTANT WEAPON… wait for it.



So while they're hunting these.




THIS is hunting them.

OK, you are a Clovis hunter. What are the requirements for your weapon? What does it have to do to be effective? The Clovis Answer?



Heres basicaly what it does.


Atlatl = Spear Thrower

-Similar technology develops around 30,000 years ago and spread worldwide.
Significant advance over spear alone.
-“Dart” replaces the spear…long, thin arrow-like projectile.
-“Atlatl” – dart throwing tool designed to increase distance and accuracy of the projectile.
-Speeds of 100 miles per hour for the dart were possible.

Parts of an Atlatl:

-Handle – 15” – 30” in length
-Notched Hook – bone/horn
-Bannerstone (Weight on back) – optional
-Wrapped Grip – various methods of retaining grip.

Why It Works:

Functioned as a lever that gave the hunter's throw more power and distance.
It extends the length of the human arm by 15” – 30” inches.
The snap of the wrist while throwing a dart from an atlatl gives you a longer lever, like adding another arm joint.


At the end of the last Ice Age, in what is now western Montana, a woolly mammoth followed a path to the river that would one day be called the Missouri. The mammoth was a lone bull, and his huge curving tusks grazed the sedge covering the mountain-fringed valley. With a flick of his trunk he sampled the breeze and snorted at the unfamiliar smell mixed with the aroma of water. Adult mammoths had no natural enemies, and this one had never before encountered the one creature that regularly killed them. As the bull paused at the strange smell, four skin-clad hunters rose from the sedge and leveled their throwing sticks. Their arms moved in an arc as each man launched a reedy projectile that sprang from the stick and covered the 40 yards in the blink of an eye. The flint points thudded deep into the hairy flanks and were followed by a second volley, then a third, which also found their marks. The mammoth trumpeted in confusion and rage, then wheeled and began a lumbering trot back up the valley, the projectiles swinging from his wounds. The hunters followed at a distance, knowing it was just a matter of time. Several of the projectiles had hit vital spots, and their quarry was soon lying dead on the tundra, a six ton mountain of meat.



Fun Atlatl Facts:

Modern record atlatl throw = 848.5 feet or nearly 3 football fields! (1995 in Aurora, Colorado)
A skilled Atlatlist can fling a 5-ounce Dart at upwards of 100 miles per hour, delivering a bigger knock-down punch than an arrow fired from a 60-pound compound bow.

WARNING! Atlatls or spearthrowers are LETHAL weapons. They are not toys. They were used to kill ancient bison and mammoth. They will make your arm effectively 2 ½ times stronger. The beginner will not have much control over where the spear lands and it will probably go further and have more force than expected. Do not use it with less respect than a bow and arrow or a gun. Use in a wide open area with nothing and no one that you do not want to accidentally kill. Use only with adult supervision. Build and use at your own risk.

Newton’s Three Laws of Awesomeness! (formerly known as Newton’s Laws of Motion)

Inertia (science) -
"In the absence of a net external force, a body either is at rest or moves with constant velocity." Lets get that one more time, in english. "Unless acted on by an outside force, an object either doesn’t move or continues in the same direction at the same speed."

Now onto Newton’s Second Law: Motion.
"The force applied to a body produces a proportional acceleration." Or "How hard you have to push or pull an object to get it to move depends on how much it weighs. Here, we'll break it down some more. Force = Mass x Acceleration. Acceleration is the change in velocity (speed) of an object either faster or slower over a given amount of time. So basicaly, how fast you want to push something depends on its weight and how much effort you put into it. "
(Hint: Write this down, it is really important!)
Newton’s Third Law: Reaction
"To every action there is an equal and opposite reaction" Or,
"When you apply force to any object, it applies the same force back on you in the form of either resistance (pushing you) or by moving from acceleration (you pushing it). "


Lets say your on a trampoline. The force you apply to it when you jump up and down is going to be reflected back on you when you land, causeing you to propell upwards. Thats the basic principle of reaction.

Universal Gravity

"Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the point masses. "


Major Huh?

Gravity exists because all objects pull toward each other. The bigger the mass, the more pull. Sun pulls Earth; Earth pulls moon; Earth pulls all objects on it toward the center of the Earth.

Gravity Explained

Objects on Earth are pulled by the largest object they are near – The Earth. Since all objects are pulled by the same force (Earth’s gravity), they are pulled the same amount.


Gravity effects objects at 9.8 meters/second2, or 32 feet/second2.

Gravity is the external force which acts up on the projectile and breaks the inertia. Gravity pulls the projectile down over time. In this way Gravity changes the speed over time. Also known as acceleration.


When gravity acts on an object in motion, it pulls the object downward over time causing it to change in velocity. A change in velocity over time is also known as?
(Hint, its acceleration)

So How much does gravity affect a projectile? .... Bueno!

Force = Mass x Acceleration due to Gravity, so instead of F = M x A, you can write it as F = M x G

Force is the amount of effort being applied
by gravity. Force is how much gravity is pulling down on the object. (Your going to want to write this one down too.)

And so it began (or, FIRST POST!!!!)

We will be discussing topics of a very mature sort. Respect and responsibility are of the upmost importance. If you cannot handle this, you need to make it known now and arrangements can be made. If you can handle this, we will have a lot of fun and a really cool science class.



Parts of the Class

There are three parts to this class:

1) The Physics: we will be discussing and working with the mathematical

and scientific concepts of physics. Mainly the six simple machines and

basic trajectory.

2) The History: you will be learning the history of weapons and warfare

because it is through these battles and people learning the physics that

we have gotten to where we are today. Without the history, you can’t

understand why the physics is important.


3) The Labs: we will be going outside, physically using the simple machines to build tools and explain the concepts. Things like giant slingshots, going to the playground, flinging stuff across long distances, and possibly working with fire to make things explode.


You are required to

Think! There are no specific answers in physics. So talk about it, explain your thoughts, ask questions even if you think they are weird or silly. The best part of physics is that you have to think outside the box, you have to experiment, you have to come up with different ways to look at the world. All great physicists and all great military leaders were renegades! Take Notes! This is less of a requirement and more of a savior for you. We will cover A LOT of information. It will be information you will have to put together yourself to come up with an explanation. Even if you are brilliant you still need to write things down, draw pictures, make connections with swirling lines in between them. These are not boring notes, these are notes so that later you can study them for tests, have them for homework, or use them in discussions in class. Pictures will help, examples will help, timelines will help…but write it down, keep it, and use it. Science classes can be incredibly cool. They can also suck. You get to choose which type this class will be. Listen and Follow Directions. This is especially important during any of the labs or excursions we take. Many of the things we will be doing involve projectiles and working with other people. If you do not listen, pay attention, and follow directions someone could get hurt.

Tentative Weekly Schedule

Monday: History and Strategy

Tuesday: Science

Wednesday: Science

Thursday: Labs, Connections, and Fun Stuff

Friday: Quiz

Quizzes will count for _________ of your grade

Participation will count for___________ of your grade



SCIENTIFIC METHOD

Question> What will happen if?...

Research> What do you know?

Hypothesis> I think…. because…

Procedure> Follow directions!

Data> Take notes

Observation> Make connections

Conclusion> Draw pictures

Any of this sounding familiar?