ADVANCED PHYSICS COURSE CHAPTER 3: TWO …gun in outer space away from any gravitational influences...

A D V A N C E D P H Y S I C S C O U R S E

C H A P T E R 3 :

T W O D I M E N S I O N A L K I N E M A T I C S

FOR HIGH SCHOOL PHYSICS CURRICULUM AND ALSO THE PREPARATION OF ACT, DSST, AND AP EXAMS

This is a complete video-based high school physics course that includes videos, labs, and hands-on learning.

You can use it as your core high school physics curriculum, or as a college-level test prep course. Either way,

you’ll find that this course will not only guide you through every step preparing for college and advanced

placement exams in the field of physics, but also give you in hands-on lab practice so you have a full and

complete education in physics. Includes text reading, exercises, lab worksheets, homework and answer keys.

BY AURORA LIPPER ∙ SUPERCHARGED SCIENCE 2017

© 2017 Supercharged Science

TABLE OF CONTENTS

Material List .............................................................................................................................................................................................................. 3

Introduction .............................................................................................................................................................................................................. 4

Vectors ........................................................................................................................................................................................................................ 5

Resultants .................................................................................................................................................................................................................. 6

Components .............................................................................................................................................................................................................. 7

Pythagorean Theorem .......................................................................................................................................................................................... 8

Relative Motion ....................................................................................................................................................................................................... 9

Boat Problem .......................................................................................................................................................................................................... 10

Crossing a River .................................................................................................................................................................................................... 11

Hot Air Balloon ...................................................................................................................................................................................................... 12

Projectile Motion .................................................................................................................................................................................................. 13

Two Dimensional Motion .................................................................................................................................................................................. 14

Using Trigonometry with Physics ................................................................................................................................................................. 16

Soccer Ball Science ............................................................................................................................................................................................... 17

P-Shooter Launcher ............................................................................................................................................................................................. 18

Pirate Problem ....................................................................................................................................................................................................... 24

Easy to Build Catapult ........................................................................................................................................................................................ 25

Two Body Problems ............................................................................................................................................................................................ 30

Advanced Catapult ............................................................................................................................................................................................... 35

Calculus and What It’s Useful For .................................................................................................................................................................. 36

Rabbit Problem ..................................................................................................................................................................................................... 37

Trebuchet ................................................................................................................................................................................................................ 38

Helpful Hints........................................................................................................................................................................................................... 45

Homeowrk Problems with Solutions ........................................................................................................................................................... 46


MATERIAL LIST

While you can do the entire course entirely on paper, it’s not really recommended since physicsis based in real-world observations and experiments! Here’s the list of materials you need inorder to complete all the experiments in this unit.

cheap mechanical pencil (you’re going to break it)

a razor with adult help

2 plastic spoons

box of popsicle sticks (large tongue-depressor size)

box of popsicle sticks (regular popsicle size)

bag of 3” x 1/4” or 3”x 1/8” rubber bands

wooden clothespins

straw

wood skewer or dowel hot glue gun

If you want to make the Trebuchet project, you’ll need to get the following supplies (Hint: watch the first

couple of minutes of the video where we go over the different parts you need.)

Tools you'll need:

Hammer

Electric drill with ¼" bit

Hot glue gun & glue sticks

Measuring tape or ruler

Hand saw & clamp (or miter box)

Scissors

Screwdriver (flathead) or wood chisel

Materials:

7 pieces of ½" x ½" x 24" pieces of wood stock

2 pieces of ¾" x 24" wood

1 piece of 3" x 24" wood

18" Wooden dowel

Screw eye

Nails

String

Clear tube

Rubber mesh

Note: wood pieces may be slightly larger or smaller than specified. Just use your best judgment when sizing.

From ½" x ½" x 24" pieces of wood stock cut:

3 pieces 5" long

2 pieces 9" long

3 pieces 3-1/2" long

4 pieces 5-1/2" long

From the dowel cut:

2 pieces 7" long

1 piece 4" long

From the 3" x 24" flat piece of wood cut:

2 pieces 3" long (one of these has a 1" square notch in it)

2 pieces 5" long

1 piece 4-1/2" long

AND...

String should be cut into 2 pieces 14-16" long

The pouch is cut from the rubber mesh and is 5" x 1-1/2"


INTRODUCTION

Objects usually travel in two or three dimensions, so now we’re going to learn how to handle more real-

world-like problems such as balls rolling down ramps, bullets fired from a rifle, baseballs thrown to home

base, and boats traveling down a river.

Be sure to take out a notebook and copy down each example problem right along with me so you take

good notes as you go along. It’s a totally different experience when you are actively involved by writing

down and working through each problem rather than passively sitting back and watching.


VECTORS

Vectors are different from scalar numbers because they also include information about direction. Velocity, acceleration, force, and displacement are all vectors. Speed, time, and mass are all scalar quantities. Acceleration can be either a scalar or a vector, although in physics it’s usually considered a vector. For example, a car traveling at 45 mph is a speed, whereas a car traveling 45 mph NW is a vector. When you draw a vector, it’s an arrow that has a head and a tail, where the head points in the direction the force is pulling or the object is moving. The coordinate system you use can be a compass (north, south, east and west) which is good for problems involving maps and geography, rectangular coordinates (x and y axes) which is good for most problems with objects traveling in two directions, or polar coordinates (radius and angle) which is good for objects that spin or rotate. We have to get really good at vectors and modeling real world problems down on paper with them, because that’s how we’ll break things down to solve for our answers. If you’re already comfortable with vectors, feel free to skip ahead to the next lesson. If you find you need to brush up or practice a little more, this section is for you. The next four videos are a review of what we’ve covered so far with vectors. If you jumped here without going through the first two sections on 1-D Kinematics or Newton’s Laws, watch these four videos now to get an overview of vector components, resultants, trigonometry, resolution, and component addition. If you’ve already worked through these, then skip down to the section on relative velocity and start there.


RESULTANTS

A resultant is the vector sum of all of the vectors, usually force vectors. You can’t just add the numbers

(magnitudes) together! You have to account for the direction that you’re pushing the box in. Here’s what

you need to know about vector diagrams and how to add vectors together:


COMPONENTS

A vector in two dimensions has components in both directions. Here’s how to add vectors together to get

a single resultant vector using component addition as well as trigonometry (the law of cosines and the

law of sines):


PYTHAGOREAN THEOREM

Vectors can be added together using the Pythagorean theorem if they are at right angles with each other

(which components always are). Here’s more practice is how to do both rectangular and polar coordinate

system components of a vector:


RELATIVE MOTION

deals with problems where one object moves with respect to another. For example, an airplane might be traveling at 300 knots according to its airspeed indicator, but since it has a 20 knot headwind, the speed you see the airplane traveling at is actually 280 knots. You’ve seen this in action if you’ve ever noticed a bird flapping its wings but not moving forward on a really windy day. In that case, the velocity of the wind is equal and opposite to the bird’s velocity, so it looks like the bird’s not moving. But what if the airplane encounters a crosswind? Something that’s not straight-on light a head or tail wind? Here’s how you break it down with vectors:


BOAT PROBLEM

These types of problems aren’t limited to airplanes, though. Have you ever gone in a boat and drifted off

course? Here’s what was happening from a physics point of view:


CROSSING A RIVER

These types of boat problems usually ask for the following information to be calculated: what is the

resultant velocity of the boat, how much time does it take to cross the river, and what distance does the

boat drift off course due to the wind? Let’s practice this type of problem again so you really can get the

hang of it.


HOT AIR BALLOON

Where else might you encounter this type of problem in the real world? Air balloons! A hot air balloon is

pretty much at the mercy of the winds, so it’s easy to calculate the component forces and velocities to

determine the path of travel. Let’s try one…

Wow! If you followed all that, you have a good working understanding of how to use math (like vectors

and equations) to solve real world problems! Don’t forget – the most important thing you do is READING

and UNDERSTANDING the problem. Don’t get hooked by shiny equations and spiffy calculations, when

sometimes the answer is as simple as dividing one number by another. I can’t tell you how many students

make this physics stuff way harder than it has to be because they’re sure they have to use fancy stuff to

get the right answer. They waste time they could have spend doing fun stuff (like science experiments!)

struggling over getting equations to fit together without understanding what those equations represent

in the first place.


PROJECTILE MOTION

If a particle moves in only one dimension, like a train on as straight track, it’s easy to answer the question about where it is because there’s only one component to it: “13m North” or “-3.6 feet.” It’s a single number with units and a positive or negative sign… that’s it. Pretty simple, right? Well, the truth is that most objects move in two or three dimensions, and so we need more information to tell us where that object is, so we use vectors. We’re going to focus on objects moving in a two dimensional plane.


TWO DIMENSIONAL MOTION

We’re going to study particles (or projectiles) that move in two dimensions. This can be a cannon ball after being fired, a baseball after being thrown, a golf ball after being hit, a soccer ball after being kicked, or any other situation you can think of where an object is under the influence of only gravity after the initial force applied to move the object. (Usually we ignore wind resistance when we do these types of problems.) The FBD of projectiles is simply a downward pointing arrow to indicate the weight. If it looks strange to have a force not in the direction of the object’s travel path, just remember that a force isn’t needed to sustain motion… it’s actually the opposite! Objects stop moving because of the forces applied to it. The FBD are always a snapshot of the forces acting on the object in that moment. The object can be moving in one direction and the force acting in another. A projectile is a particle that is only experiencing gravity, and in most cases, gravity is only acting in one direction. Gravity doesn’t influence the horizontal motion (if we accounted for air resistance, then there would be a force in this direction as well), only the vertical motion. That’s why the ball falls to the ground when you throw it. This means that a bullet fired horizontally from a gun experiences a constant horizontal velocity and a

downward vertical acceleration. A bullet fired from a gun pointed up at a 45 degree angle also

experiences a constant horizontal velocity and a downward vertical acceleration. A bullet fired from a

gun in outer space away from any gravitational influences would travel up at a 45 degree path away from

the gun and experience constant horizontal and vertical velocity.

The path a projectile makes is parabolic, meaning that it follows the shape of a parabola. The horizontal motion of the projectile is independent of the vertical motion. You’ll need to think about each component as separate and independent.


2-D Kinematic Equations

vx = vxo + axt vy = vyo + ayt x = xo + vxot + 1/2 axt2 y = yo + vyot + 1/2 ayt2

vx2 = vxo2 + 2 ax (x – xo) vy2 = vyo2 + 2 ay (y – yo) We can transform the above equations into a set of equations specifically for projectile motion by setting the acceleration in the x direction equal to zero for constant velocity (ax = 0) and setting the acceleration in the y direction equal to gravity (ay = -g) and rewrite to get: vx = vxo vy = vyo – gt

x = xo + vxot y = yo + vyot – 1/2 gt2

vy2 = vyo2 – 2 g (y – yo)


USING TRIGONOMETRY WITH PHYSICS

Now let’s do a set of physics problems so you can really see how to solve these. The first one shows you

how to not only calculate an angle buried in a trig function, but also that you don’t need fancy equations

to solve a problem and that you really have to understand what the problem is asking for, so you don’t

waste time calculating stuff you don’t need.


SOCCER BALL SCIENCE

This problem will show you how a soccer ball can also be a projectile, and how by knowing a couple of

simple things, you can find out everything you need about the problem, including how far and how high

the ball traveled in addition to its time of flight.


P-SHOOTER LAUNCHER

We’re going to make a projectile launcher from a rubber band and a pencil as a study break, since you’ve

been working so hard! Here’s what you do:


P-Shooters

Introduction: P-Shooters are a fun, simple, and sneaky way of launching tiny objects. You can turn some simple household items into a device that shoots tiny objects at high speeds! How does this work? These small P-Shooters use the power of elastic potential energy to launch small projectiles. This is possible because of conservation of energy. The elastic potential energy is released and converted into kinetic energy in the projectile!

Materials:

a cheap mechanical pencil two rubber bands

a razor with adult help

stopwatch

Procedure:

Making the P-Shooter:

1. Take your mechanical pencil and carefully break off the writing tip to expose the main tube. 2. Remove all the springs, lead, and any other pieces and discard them. 3. Break off the pocket holder clip, you should now be left with a tube that can move freely inside

another tube. 4. Take the razor and carefully make a cut all the way across the tip of the eraser. 5. Now take one rubber band and slide it inside the cut you just made so it sits securely in the eraser. 6. Stretch the rubber band most of the way down the pencil. Take your second rubber band and secure

the first stretched rubber band in place by wrapping it around and around the pencil, until it holds the first rubber band tightly.

7. Get some small projectiles, like a wadded up piece of paper, and see how far it will fly!

Now that you've got your P-Shooter working, let's do some physics. Let's calculate the vertical speed of the projectile if it's launched at 45°. Grab your stopwatch and a good projectile. Load the P-Shooter, and hold it at a 45° angle with the tip as close to a table top as possible, like the picture below. Start the stopwatch and release the P-Shooter at the exact same time. When the projectile hits the table top, stop the stopwatch and write down the time in the table on the next page. Repeat this 5 times for accuracy.


Once you've measured the times for five different flights let's calculate the initial vertical speed component. To do this we need to use this kinematic equation to describe the first half of the total motion: v f = vi + a t. For vf, we will use 0 m/s, because at the peak height of the projectile, the vertical speed component is 0. The

acceleration, a, is equal to gravity pulling the object down (-9.8 m/s2). For time, t, we will use ½ of the total

time measured, because this is how long it takes to reach the peak height. For example, if we measured 2.4 seconds for the total flight time, here's how we would find the initial vertical speed:

Calculation for Trial #1:

Once you've filled out the first 5 rows of the table, find the average time and average initial vertical speed. Plot this on the velocity vs time graph below. Make sure to add in values for your axes! (Note: the initial velocity will be a positive value. It will slow down until it reaches the maximum height, then it will come back down with a negative velocity)

Trial # Time (seconds) Initial Vertical Speed (m/s)

1

2

3

4

5

Average


Problems:

1. If a gun is fired straight up in the air, and the bullet takes 84 seconds to fall back to Earth, what was

the initial velocity of the bullet?


2. A skier goes off a jump with a horizontal velocity of 10 m/s, and a vertical velocity of 7 m/s. How long will

it take him to land back down?

3. Another skier goes off the same jump, but has a horizontal velocity of 13 m/s, and a vertical velocity of 4

m/s. How far will the skier travel (horizontally) before she lands?


Answers:

1. 412 m/s 2. 1.4 seconds 3. 10.6 m


PIRATE PROBLEM

Okay now, back to work! Here’s a neat problem involving a pirate ship and a cannon ball. I seriously

doubt pirates would be able to calculate this kind of problem when being fired at by a fortress, but you

might have a captain that had a good sense based on experience of how far and fast that cannon ball

could travel.


EASY TO BUILD CATAPULT

Now physics isn’t all math and equations. We’re going to build a projectile launcher right now! Here’s a

very simple catapult you can make right now:


Catapults

Overview: Turns out the ancient people could teach us a thing or two about energy when they laid

siege to anenemy town. Although we won’t do this today, we will explore some of the important

physics concepts that they have to teach us.

What to Learn: Energy can be found in many forms. Identify what kinds and where each type of energy areworking in this experiment, and you’ll be ready to move on.

Materials

9 tongue-depressor size popsicle sticks

four rubber bands

one plastic spoon

ping pong ball or wadded-up ball of aluminum foil (or something lightweight to toss, like a marshmallow)

hot glue gun with glue sticks

Lab Time

1. Stack seven popsicle sticks and secure them together with rubber bands. Twist them around a few times so they stay securely. Do this on each end.

2. Grab two more popsicle sticks, stack them, and secure one end with a rubber band. The other end will stay open. We’ll slide our fulcrum into the open end.

3. Slide the open end over the seven stacked sticks, and secure the whole thing by crossing a rubber band over the end of the two stacks.

4. Attach the spoon to the end of the upward-facing stick with hot glue or an extra rubber band.

5. Take the aluminum foil and scrunch it into a ball. Place the ball on the spoon, press it down, and release!


Catapult Observations

1. What part of the catapult stores the most potential energy? Why is this?

2. Where is the kinetic energy transferred to in this catapult?

3. How would you make a catapult’s projectile travel farther? Explain.

Troubleshooting: These simple catapults are quick and easy versions of the real thing, using a fulcrum

instead of a spring so kids don’t knock their teeth out. After making the first model, encourage kids to

make their own “improvements” by handing them additional popsicle sticks, spoons, and glue sticks

(for the hot glue guns).

If they get stuck, you can show them how to vary their models: glue a second (or third, fourth or fifth)

spoon onto the first spoon for multi-ammunition throws, increase the number of popsicle sticks in the

fulcrum from 7 to 13 (or more?), and/or use additional sticks to lengthen the lever arm. Use ping pong

balls as ammo and build a fort from sheets, pillows, and the backside of the couch.

Reading

We’re utilizing the “springy-ness” in the popsicle stick to fling the ball around the room. By moving the

fulcrum as far from the ball launch pad as possible (on the catapult), you get a greater distance to press

down and release the projectile. (The fulcrum is the spot where a lever moves one way or the other –

for example, the horizontal bar on which a seesaw “sees” and “saws”.)


Exercises Answer the questions below:

1. How is gravity related to kinetic energy?

a. Gravity creates kinetic energy in all systems.

b. Gravity explains how potential energy is created.

c. Gravity pulls an object and helps its potential energy convert into kinetic energy.

d. None of the above

2. If you could use your catapult to launch your ball of foil into orbit, how high would it have to go?

a. Above the atmosphere

b. High enough to slingshot around the moon

c. High enough so that when it falls, the earth curves away from it

d. High enough so that it is suspended in empty space

3. Where is potential energy the greatest on the catapult?


Answers to Exercises: Catapults

1. How is gravity related to kinetic energy? (Gravity pulls an object and helps its potential energy convert into kinetic energy.)

2. If you could use your catapult to launch your ball of foil into orbit, how high would it have to go? (high enough so that when it falls back down the earth is already curving away)

3. Where is potential energy the greatest on the catapult? (when the spoon is pressed down all the way)


TWO BODY PROBLEMS

Two body problems are more common than you might think. Here’s a two-dimensional two body

problem that is a good review of Newton’s Second Law that also includes friction calculations.

Wa-hoo! You’ve made it through a seriously hard set of physics problems. Honestly, a lot of students get

stuck in 2D equations, making them impossibly hard to solve. But if you follow the steps slowly and read

the problem carefully and understand what it’s asking for, your path should be relatively easy and

smooth sailing.


Catapults

Overview: Turns out the ancient people could teach us a thing or two about energy when they laid

siege to anenemy town. Although we won’t do this today, we will explore some of the important

physics concepts that they have to teach us.

What to Learn: Energy can be found in many forms. Identify what kinds and where each type of energy areworking in this experiment, and you’ll be ready to move on.

Materials

9 tongue-depressor size popsicle sticks

four rubber bands

one plastic spoon

ping pong ball or wadded-up ball of aluminum foil (or something lightweight to toss, like a marshmallow)

hot glue gun with glue sticks

Lab Time

1. Stack seven popsicle sticks and secure them together with rubber bands. Twist them around a few times so they stay securely. Do this on each end.

2. Grab two more popsicle sticks, stack them, and secure one end with a rubber band. The other end will stay open. We’ll slide our fulcrum into the open end.

3. Slide the open end over the seven stacked sticks, and secure the whole thing by crossing a rubber band over the end of the two stacks.

4. Attach the spoon to the end of the upward-facing stick with hot glue or an extra rubber band.

5. Take the aluminum foil and scrunch it into a ball. Place the ball on the spoon, press it down, and release!


Catapult Observations

1. What part of the catapult stores the most potential energy? Why is this?

2. Where is the kinetic energy transferred to in this catapult?

3. How would you make a catapult’s projectile travel farther? Explain.

Troubleshooting: These simple catapults are quick and easy versions of the real thing, using a fulcrum

instead of a spring so kids don’t knock their teeth out. After making the first model, encourage kids to

make their own “improvements” by handing them additional popsicle sticks, spoons, and glue sticks (for

the hot glue guns).

If they get stuck, you can show them how to vary their models: glue a second (or third, fourth or fifth)

spoon onto the first spoon for multi-ammunition throws, increase the number of popsicle sticks in the

fulcrum from 7 to 13 (or more?), and/or use additional sticks to lengthen the lever arm. Use ping pong

balls as ammo and build a fort from sheets, pillows, and the backside of the couch.

Reading

We’re utilizing the “springy-ness” in the popsicle stick to fling the ball around the room. By moving the

fulcrum as far from the ball launch pad as possible (on the catapult), you get a greater distance to press

down and release the projectile. (The fulcrum is the spot where a lever moves one way or the other – for

example, the horizontal bar on which a seesaw “sees” and “saws”.)


Exercises Answer the questions below:

1. How is gravity related to kinetic energy?

a. Gravity creates kinetic energy in all systems.

b. Gravity explains how potential energy is created.

c. Gravity pulls an object and helps its potential energy convert into kinetic energy.

d. None of the above

2. If you could use your catapult to launch your ball of foil into orbit, how high would it have to go?

a. Above the atmosphere

b. High enough to slingshot around the moon

c. High enough so that when it falls, the earth curves away from it

d. High enough so that it is suspended in empty space

3. Where is potential energy the greatest on the catapult?


Answers to Exercises: Catapults

1. How is gravity related to kinetic energy? (Gravity pulls an object and helps its potential energy convert into kinetic energy.)

2. If you could use your catapult to launch your ball of foil into orbit, how high would it have to go? (high enough so that when it falls back down the earth is already curving away)

3. Where is potential energy the greatest on the catapult? (when the spoon is pressed down all the way)


ADVANCED CATAPULT

Since you’ve worked so hard, I thought you’d enjoy making a marshmallow launcher just for fun. You can

calculate the horizontal and vertical acceleration based on the time of flight, you can also figure out the

initial speed based on how far it went, or you can just make it and have fun with it. Here it is:


CALCULUS AND WHAT IT’S USEFUL FOR

So now let’s look ahead and sneak a peek into your future. Are you nervous about taking Calculus? Or if

you have, have you wondered what Calculus could possibly be useful for? Here’s a two part video that

shows you what Calculus is (and will even have you doing it before the end of the second video!) and how

it’s used all the time in physics. Sir Isaac Newton was so frustrated that he couldn’t solve his physics

problems with the math that was already developed at that time (algebra) that he set them aside to

invent a branch of mathematics that could support his work in science, and that’s where Calculus came

from. Here’s how we use it today in physics…


RABBIT PROBLEM

Now that you know what functions are, here’s how to solve the rabbit problem:


TREBUCHET

If you’ve ever wanted to make a real working trebuchet out of simple materials you’d find at the

hardware store, here’s complete instructions. This project will take about a weekend with an experienced

adult who’s good at using tools.


Trebuchet

Introduction: For ages, people have been hurling rocks, sticks, and other objects through the air. The

trebuchet came around during the Middle Ages as a way to break through the massive defenses of castles

and cities. It’s basically a gigantic sling that uses a lever arm to quickly speed up the rocks before letting go.

Trebuchets are really levers in action. You’ll find a fulcrum carefully positioned so that a small motion near

the weight transforms into a huge swinging motion near the sling. Some mis-named trebuchets are really

‘torsion engines’, and you can tell the difference because the torsion engine uses the energy stored in twisted

rope or twine (or animal sinew) to launch objects, whereas true trebuchets use heavy counterweights.

This is a serious wood construction project, so the materials list is extensive. The process of constructing

your trebuchet is described in detail during the video.

Materials:

Tools:

Hammer

Electric drill with ¼” bit Hot glue gun

Measuring tape or ruler

Hand saw & clamp (or miter box) Scissors

Screwdriver (flathead) Stopwatch or timer

Materials:

7 pieces of ½” x ½” x 24″ pieces of wood stock

2 pieces of ¾” x 24″ wood

1 piece of 3″ x 24″ wood

18″ Wooden dowel Screw eye

Nails

String

Clear tube

Rubber mesh

From ½” x ½” x 24” pieces cut:

3 pieces 5″ long

2 pieces 9″ long

3 pieces 3-1/2″ long

4 pieces 5-1/2″ long


From the dowel cut:

2 pieces 7” long

1 piece 4” long

From the 3″ x 24″ flat piece of wood cut:

2 pieces 3″ long (one of these has a 1″ square notch in it) 2 pieces 5″ long

1 piece 4-1/2″ long

AND...

String should be cut into 2 pieces 14-16″ long

The pouch is cut from the rubber mesh and is 5″ x 1-1/2″

Once you've got your stock cut, materials organized, and tools ready, watch the video online and

follow closely to start building your trebuchet.

Once you've got your trebuchet working nicely, let’s get down to business. There's a whole lot of physics

going on when you wind up your trebuchet and release it. You're taking the kinetic energy of your hand

winding up the lever, then taking the now-potential energy of the lever and converting it back into kinetic

energy in the projectile.

Let's start looking closer at some of the aspects of the trebuchet's motion. Let's start by doing some trials and

taking some important measurements. Set up your trebuchet so that it will hit a wall or surface that can

safely take a beating. Next, measure the distance from the wall to your trebuchet. Fill out the table below

with this distance in the corresponding column.

Once you've got your trebuchet set up at a known distance, grab your timer. Load the trebuchet, let it go,

and start the timer once the projectile is released form its pouch. Stop the timer when it hits the wall, and

record the time in the corresponding row and column in the table below.


Once you've filled out the measured x-distances and times for three trials, let's calculate the launch velocity,

as well as the launch acceleration. To calculate the launch velocity, use the following kinematic equation

with your xi = 0 m/s, and xf equal to the measured x-distance in meters:

vx = (xf – xi) / t

Calculate the launch velocities for each of your trials, and fill in the corresponding column.

Now, to solve for the launch acceleration we need to know one more piece of information. How far does the

projectile travel from it's initial rest position to its release position? We can't simply take a ruler and

measure from point A to point B, because we need to know the distance of the path that it traveled. We can

approximate this by modeling the path of the projectile as about half the circumference of a circle. The

equation for the circumference of half of a circle is:

½ C = ½ 2 π r

To get the radius of this circle, measure the distance from the pivot point of your trebuchet to the center of the

pouch (when the string is tight). Once you plug in this value to the equation above, you will have an approximate

value for the distance the projectile traveled. Now we can calculate the acceleration of the launch by using the

following equation, where vi = 0 m/s, vf is equal to the launch velocity calculated above in m/s, Δx is the distance

the projectile traveled in meters, and a is the acceleration we are solving for in m/s2:

vf2 = vi

2 + 2a Δx

a = vf2 / (2 Δx)

Calculate the launch accelerations of each of your three trials and fill in the remaining column in the table.


Problems:

1. A large scale trebuchet launches a projectile at 35 mph. If it's path length is 8.6 meters,

what acceleration does the projectile undergo during launch?

2. If the trebuchet is twice as big (path length is 17.2 m), and it launched the projectile with the

same speed, what is the new acceleration.


3. If the path length is again 8.6 meters, but the projectile is now launched at 70 mph, what is

the acceleration?


Answers:

1. 14.24 m/s2

2. 7.12 m/s2

3. 56.92 m/s2


HELPFUL HINTS

Physics can really trip you up if you’re not careful! You can to remember all kinds of things, including

triangles, significant figures, vectors, units, and so much more. Here’s a video on some helpful hints to

keep in mind as you go along:

Yay! You’ve completed this section! Now it’s your turn to solve your own set of physics problems:


HOMEWORK PROBLEMS WITH SOLUTIONS

On the following pages is the homework assignment for this unit. When you’ve completed all the videos from this unit, turn to the next page for the homework assignment. Do your best to work through as many problems as you can. When you finish, grade your own assignment so you can see how much you’ve learned and feel confident and proud of your achievement! If there are any holes in your understanding, go back and watch the videos again to make sure you’re comfortable with the content before moving onto the next unit. Don’t worry too much about mistakes at this point. Just work through the problems again and be totally amazed at how much you’re learning. If you’re scoring or keeping a grade-type of record for homework assignments, here’s my personal philosophy on using such a scoring mechanism for a course like this: It’s more advantageous to assign a “pass” or “incomplete” score to yourself when scoring your homework assignment instead of a grade or “percent correct” score (like a 85%, or B) simply because students learn faster and more effectively when they build on their successes instead of focusing on their failures. While working through the course, ask a friend or parent to point to three questions you solved correctly and ask you why or how you solved it. Any problems you didn’t solve correctly simply mean that you’ll need to go back and work on them until you feel confident you could handle them when they pop up again in the future.

Advanced Physics 2-D Kinematics

© 2014 Supercharged Science www.ScienceLearningSpace.com Page 1

Student Worksheet for Two Dimensional Kinematics After you’ve worked through the sample problems in the videos, you can work out the problems below to practice

doing this yourself. Answers are given on the last page.

Kinematic Equations: Projectile Motion Equations:

vx = vxo + axt vy = vyo + ayt

x = xo + vxot + 1/2 axt2 y = yo + vyot + 1/2 ayt2

vx2 = vxo

2 + 2 ax (x - xo) vy

2 = vyo2 + 2 ay (y - yo)

vx = vxo vy = vyo - gt

x = xo + vxot y = yo + vyot - 1/2 gt2

vy2 = vyo

2 - 2 g (y - yo)

Where: t=time, d=displacement, x=position, v=velocity, a=acceleration, o=initial

Practice Problems:

1. A car drives 150 km east from city A to B in 45 minutes and then 300 km south from city B to C in 1.5 hours.

What is the average velocity vector for the trip?



2. If a ball is thrown horizontally with a speed of 65 mph, how far will it fall while traveling 90 ft of horizontal

distance?

3. A sniper is shooting at a target 1 km away horizontally. The bullet hits the target 50 cm below the aiming

point. What is the bullet’s time of flight?



4. What is the velocity as the bullet leaves the sniper’s gun in question 3 above?

5. A football is thrown with an initial velocity of 73 ft/s at an angle of 45o above the horizontal. What is the

velocity at 3 seconds after the ball is thrown?



6. You throw a water balloon off a roof at an initial velocity of 10 m/s at an angle of 40O below the horizontal.

Find the displacement 2 seconds later.

7. A missile is launched at is 55 degrees up from the horizontal. What is the initial speed for the missile to land

10 km horizontally away and 2 km vertically lower than the launch point?



8. A soccer player kicks the ball so that it will have a time of flight (hang time) of 3 seconds and land 30 yards

away. If the ball leaves the player’s foot 3 feet above the ground, what is the initial velocity?

9. What is the maximum vertical height to which a golfer can hit a ball if he can hit it a maximum distance of

300 yards?



10. The launch speed of a projectile is three times the speed it has at its maximum height. What is the elevation

angle at launch?

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MECHANICS ELECTRICITY

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m = coefficient of friction q = angle r = density t = torque w = angular speed

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WAVES

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GEOMETRY AND TRIGONOMETRY

Rectangle A bh�

Triangle 12

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Circle 2A p�

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r

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Cylinder

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CONSTANTS AND CONVERSION FACTORS

Proton mass, 271.67 10 kgpm ��

Neutron mass, 271.67 10 kgnm ��

Electron mass, 319.11 10 kgem ��

Avogadro’s number, 23 -10 6.02 10 molN � �

Universal gas constant, 8.31 J (mol K)R � �

Boltzmann’s constant, 231.38 10 J KBk ��

Electron charge magnitude, 191.60 10 Ce ��

1 electron volt, 191 eV 1.60 10 J�� Speed of light, 83.00 10 m sc � �

Universal gravitational constant,

11 3 26.67 10 m kg sG ��

Acceleration due to gravityat Earth’s surface,

29.8 m sg �

1 unified atomic mass unit, 27 21 u 1.66 10 kg 931 MeV c�� Planck’s constant, 34 156.63 10 J s 4.14 10 eV sh ��

25 31.99 10 J m 1.24 10 eV nmhc ��

Vacuum permittivity, 12 2 20 8.85 10 C N me ��

Coulomb’s law constant, 9 201 4 9.0 10 N m Ck pe� � � �

AVacuum permeability, 70 4 10 (T m)m p ��

Magnetic constant, 70 4 1 10 (T m)k m p ��

5 1 atmosphere pressure, 5 21 atm 1.0 10 N m 1.0 10 Pa� � � �

UNIT SYMBOLS

meter, m kilogram, kgsecond, sampere, Akelvin, K

mole, mol hertz, Hz

newton, Npascal, Pajoule, J

watt, W coulomb, C

volt, Vohm,

henry, H

farad, F tesla, T

degree Celsius, C� W electron volt, eV

2

A

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PREFIXES Factor Prefix Symbol

1012 tera T

109 giga G

106 mega M

103 kilo k

10�2 centi c

10�3 milli m

10�6 micro m

10�9 nano n

10�12 pico p

VALUES OF TRIGONOMETRIC FUNCTIONS FOR COMMON ANGLES

q �0

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53 60� 90�

sinq 0 1 2 3 5 2 2 4 5 3 2 1

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The following conventions are used in this exam. I. The frame of reference of any problem is assumed to be inertial unless

otherwise stated. II. In all situations, positive work is defined as work done on a system.

III. The direction of current is conventional current: the direction in whichpositive charge would drift.

IV. Assume all batteries and meters are ideal unless otherwise stated.V. Assume edge effects for the electric field of a parallel plate capacitor

unless otherwise stated.

VI. For any isolated electrically charged object, the electric potential isdefined as zero at infinite distance from the charged object.

MECHANICS ELECTRICITY AND MAGNETISM

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FLUID MECHANICS AND THERMAL PHYSICS

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system by heating T = temperature t = time U = internal energy V = volume v = speed W = work done on a system y = height�r = density

mV

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GEOMETRY AND TRIGONOMETRY

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Rectangle A � bh

Triangle 12

A b� h

Circle 2A rp�

2C rp�

Rectangular solid V w� � h

r

Cylinder 2V rp� �

22S rp p� �� 2

Sphere 34

3V p� r

24S rp�

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sin ac

q �

cos bc

q �

tan ab

q �

c a

b90�q

ADVANCED PHYSICS COURSE CHAPTER 3: TWO …gun in outer space away from any gravitational influences...

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Transcript of ADVANCED PHYSICS COURSE CHAPTER 3: TWO …gun in outer space away from any gravitational influences...