AP-B Physics



AP Physics 1 Summer Assignment 2019

Name: __________________________________________________Date: __________________

Congratulations! You have decided to take AP Physics B. You are making extremely good use of your high school years by taking a class where you will actually learn something as well as provide you with a stiff challenge. This summer homework will allow us to start on the Physics subject matter immediately when school begins. If you have any questions, difficulties, or problems please email me at aguaty@dadeschools.edu.

In addition to the science concepts, physics can seem like a course in applied mathematics. AP Physics B requires proficiency in algebra, trigonometry, and geometry. In this assignment, you will have the opportunity to review these concepts. If you do not have a problem with these concepts, you will have no problem with the math necessary to be successful in this class. For most people, the challenge in Physics is in knowing when to use the necessary math skills, not the math skills themselves. You will have a lot of opportunities to develop the skills necessary to “do physics” successfully during the course of the year.

In addition to the math review, this assignment will cover essentially the all the material from the entire first two chapters of our textbook. The expectation is that you will arrive having fully completed this assignment and have an understanding of the material. Therefore, the due date for summer assignment will be the first day of school. We will have a quiz over this material within the first three days of school. If you are having a hard time understanding the problems or instructions, please email me at the above email address and I will get back to you.

To be successful on this summer assignment, you should do the following: 1) Read every single word in it, don’t skip around. 2) Study the material. Make study cards, highlight important points, and engage your brain with the material. 3) Try doing the example problems on your own. Often in Physics there are multiple ways to solve a specific problem. I’ll generally show you only one way, but to become proficient at this stuff you should see if you can figure out other ways to a solution. This is not a math class; I will encourage you and challenge you to find multiple methods of solving physical situations. 4) Do not rely solely on the information presented in this packet. The internet has a HUGE amount of information that will help you. There are tutorials and videos everywhere that will help you understand the concepts presented in this packet. Here are a few websites that you can use to help you understand the Physics topics presented in this assignment:

- Great tutorial site. Go through all six lessons to get a really good grasp of all the topics of our first unit.

- Another great tutorial site presented in a concept map format. Click on the “mechanics” bubble to access information necessary for this assignment.

- A series of videos on many, many subjects. This site should become an indispensable resource for you throughout the year.

. A great tutorial site.

See you in August! (

Section I: Math Review

1. Being able to manipulate formulae to solve for a variable is an extremely important skill in Physics. It is done to isolate a single variable to make problem solving easier. You will be accustomed to solving problems in this manner by the end of the course. The formulae below are a few of the actual ones we will be learning during the course. Your task is to manipulate the variables algebraically and solve for the variable indicated.

a. [pic] ______________

b. [pic] ,m = ______________

c. [pic] [pic]______________

d. [pic] ______________

e. [pic] , I= _______________

f. [pic] _______________

g. [pic] _______________

h. [pic] [pic] ________

2. The geometry skills necessary in Physics involve being able to calculate angles, find lengths of lines, and understand basic geometric terms. Solve the following geometric problems using the figures provided.

a. In figure A, line B touches the circle at a single point. Line A extends through the center of the circle.

i. What term can be used to describe line B in reference to the circle? _______________

ii. How large is the angle between lines A and B? _______________

iii. If the radius of the circle is 5.5 cm, what is the circumference in meters? ____________

iv. If the radius of the circle is 5.5 cm, what is the area in square meters? ______________

b. In figure B, what is the measure of angle C? ______________

c. In figure C, what is the measure of angle ( ? _____________

d. The diagram above shows an object sitting on a ramp. A coordinate axis has been included for reference and is tilted along the x-axis. The diagram is not drawn to scale. This is a common diagram in Physics when dealing with objects on sloped surfaces. How large is angle(?

___________________

e. One of the first concepts we will be dealing with involves using graphs to describe the motion of objects. One aspect of these graphs that you will become accustomed to hearing is referred to as “the area under the curve.” This refers to the area of the geometric shape created by the graph. Using this graph, calculate the following:

i. What is the area under the curve? _____________

ii. What is the slope of section A? _______________

iii. What is the slope of section B? _______________

3. Using the generic triangle to the right, Right Triangle Trigonometry and Pythagorean Theorem solve the following. Your calculator must be in degree mode.

a. ( = 55o and c = 32 m, solve for a and b.

_______________ _______________

b. ( = 45o and a = 15 m/s, solve for b & c.

_______________ _______________

c. b = 17.8 m and ( = 65o, solve for a & c.

______________ ________________

This concludes the math review portion of the summer assignment. The math concepts presented here pretty much sum up the level of math we will be dealing with throughout the course. If you can adequately do some algebra, know your basic geometry, and can do some right triangle trigonometry, you can do the math in Physics. Pretty easy stuff isn’t it? Most people don’t have much difficulty with the math of physics. The challenge lies in being able to take a situation, apply physical principles to it, and use mathematics to bring out the details of it.

Section II: First-Order Stuff

As an AP Physics student, you will be privileged to solve many problems – some of them quite involved and complicated. You will become a master problem solver through the course of the year. To help get you started, I have provided you with some helpful problem solving hints. You will see and hear these multiple times throughout the course.

Physics Problem Solving Tips

1. Read the problem carefully at least twice.

2. Draw a diagram of the basic situation with labels. (The diagram doesn’t have to be artistic!)

3. Decide on what direction will be positive. Decide on where to place the origin of your coordinate system. (You know the x- and y-axis?)

4. Imagine a movie in your mind of what happens. What does your common sense tell you about what is going to happen? (This is probably the most useful step in this process)

5. Identify the basic physics principles involved; list the known and unknowns of the problem.

6. Determine what information is important and what information is extraneous (unnecessary).

7. Draw a free body diagram (if appropriate – more on what a free body diagram is later).

8. Write down or develop the equation(s) needed.

9. Substitute the given values into the equations you developed.

10. Do the calculations.

11. Now check that all numbers have proper units.

12. Reflect upon your work. Ask yourself these questions:

• Are the units the proper ones?

• Is the answer reasonable – does it make sense?

• Is a plus or minus sign proper or meaningful?

• Are there other ways to solve this that I could use to check my answer?

13. Draw a box or a circle around your answer.

Your work should be written as a logically ordered series of discrete, clearly delineated steps. This will allow another person to follow your method of problem solving. In other words, don’t “shotgun” your work by writing it down in a helter-skelter fashion where it’s hard to follow what you’ve done.

Sample Problem and Solution Format (A table like the one below is not necessary. The example below is written horizontally to save space. You may choose to list your steps horizontally or vertically. The first step – stating the given information – is optional, but it is a good idea for first-time physics students.)

|State the |Write the |Plug in Numbers |Do the Math |Round to the |

|Given |Equation |with Units |(include units |Correct |

|Information |(Rearrange as |(Notice that units will |in the |Number of Sig. |

| |Necessary) |cancel/combine to get |answer) |Figs. and Box in |

| | |the appropriate units | |Your Final |

| | |for your answer.) | |Answer |

|v = 1.8 m/s |v = d ⋄ d = vt |d = (1.8 m/s)(235 s) |= 423 m |= 420 m |

|t = 235 s |t | | | |

|d = ? | | | | |

Unfortunately, before we can really get into the exciting world of physics, we got to get some basics out of the way. You’ve probably taken chemistry, algebra, possibly pre-calculus or AP calculus, maybe even general physics, so you are already familiar with how to do math and science. Much of the material in this section you already know or were at least exposed to. However, some of it will be brand new information.

The Units of Physics

We will use the SI (International System of Measurement) system exclusively. The SI system is also known as the metric system. The standard units are:

|Quantity |Unit |Abbreviation |

|Mass |Kilogram |kg |

|Time |Seconds |s |

|Length |Meter |m |

All the other units we will use are pretty much derived from these three basic units.

To measure large or small things with these units, prefixes are added which alter the value of the unit. Here are the key prefixes that we will use:

It will be important for you to understand how to convert prefixed units to standard units. In most cases, it’s moving a decimal left or right. In others, it becomes a bit more complicated and we’ll take a look at it on the next page. In all our problems we will always convert to standard units before solving.

Dimensional Analysis

Dimensional analysis is a system for unit conversions. It comes in handy whenever you have to do an involved, complicated unit conversion. Simple conversions like centimeters to meters, however, you can just do in your head. The method is taught in chemistry, so you may be familiar with it. The key idea here is that units are treated like algebra symbols – you can multiply them by each other, divide with them, and often cancel them out.

• Example 1: There are 5280 ft in one mile, 3.281 feet in one meter, and 3600 seconds in one hour. Convert 75.0 miles/hour to m/s.

[pic]

Note the diagonal line through some of the units – these are the units that cancelled. Make sure that you always cancel units where required. Dimensional analysis is always a good check on whether you’ve set up a problem properly. Usually if the units work out, then you’re solution method is a proper one.

Significant Figures and Rounding

At this point you might be wondering about the importance of significant figures in AP Physics. In Chemistry, you probably had to deal with significant figures and make sure you followed all those significant figure rules and whatnot. In Physics, we don’t have to worry too much about it. You won’t be asked to ensure that all your answers have the correct number of significant figures. However, you will be asked to make sure your answers look reasonable and are at least rounded appropriately. If you get a calculator answer that looks like this: 34.56794432, you would report it as 34.57 or 34.6, but not the whole thing.

Basic Physics Definitions (This will likely be brand new information for you)

Here are some basic definitions of terms we will use constantly throughout the course. You need to make sure you understand them.

Kinematics: The study of motion independent of its causes. This simply means that you study movement but you don’t worry about what made the movement happen.

Dynamics: The study of motion and its causes. (Motion is caused by forces).

Scalar: A quantity (something you can measure) that has magnitude (a size) only. Mass, temperature, distance, and density are all examples of scalar quantities. These only have a specific size, like 10 kg for example. You get all the meaning you need out of just the number.

Vector: A quantity that has magnitude and direction. Forces, accelerations, and velocities are examples of vector quantities. A vector gives you both the size and the direction. For example, to describe the velocity of a car you would say “The car was going 35 mph west.” If all you say is, “The car was going 35 mph,” you have conveyed only information about its speed, but not how to locate the position of the car.

Section III: Kinematics Basics

So now we can start some actual Physics. In this section, you will learn about speed and velocity, be given your first physics equation and some problems, and do your first physics lab. We’ll start with a description of some important terms. After each term is the symbol used to abbreviate that term. These are used as variables in equations.

Position, x: Where an object is at a specific point in time. For example, an object can be at +5 m on the x-axis or at -7.0 m on the y-axis. We will often talk about an object’s initial position or final position. This simply refers to the position where an object starts and the position where it ends up.

Distance: A linear length. Distance is a scalar quantity. Distance is a measure of the total amount of space traveled. When we talk about the distance an object has gone, we are talking only about the total amount of space it has covered. We aren’t concerned about what direction the thing has gone.

Displacement. ∆x: A change in position. Displacement is a vector for how far an object has traveled in a straight line. Displacement goes with velocity and is measured from the origin. It is a value of how far away from the origin you are at the end of the problem. The direction of a displacement is the shortest straight line from the location at the beginning of the problem to the location at the end of the problem.

The ∆ symbol is the Greek letter delta and stands for “change in.” It is used frequently in Physics. For example, here it means “change in position.” Any time you see the ∆, you should immediately think to yourself, “I know there is a final and initial here.” In the case of displacement, you will think about there being a final position and an initial position. Mathematically, you will think:

∆x = xf - xi

• Example 2: If an object’s initial position is at +8.0 m on the x-axis and its final position is at +2.0 m on the x-axis, what is the object’s displacement?

[pic]

[pic]

[pic]

Here, the displacement is -6.0 m. This means that the object moved 6.0 m in the negative direction.

How do distance and displacement differ? Suppose you walk 20 meters along the + x axis then turn around and walk 10 meters along the – x axis. The distance traveled does not depend on direction since it is a scalar, so you walked 20 m + 10 m = 30 meters. Displacement, on the other hand, does depend on direction and cares about how far you are from the origin at the end of the problem. The origin is considered your initial position and is at 0 m. Because you walked 20 meters, then turned around and walked 10 meters, your displacement from the origin is 10 meters.

[pic]

[pic]

[pic]

Direction in Physics

Let’s take a moment to discuss direction. In Physics, we can talk about a vector acting in the positive or negative direction. By convention, positive and negative direction is as follows:

|Up or + y-axis = positive |Right or +x-axis = positive |

|Down or – y-axis = negative |Left or –x-axis = negative |

So, if an object is initially at +6.5 on the x-axis and moves 3.0 units to the left, its displacement will be negative. If it moves 3.0 units to the right, its displacement will be positive.

[pic]

Speed, vs: A measure of how fast something moves. It is a rate. Rates are quantities divided by time. In addition, speed is a scalar quantity. When you talk about speed, you only care about how fast the thing is going over a specific time interval. For example, vs = 10 m/s means that an object is going 10 meters every second. But, we do not know where it is going.

Velocity, v: The rate at which displacement changes over time. Velocity is a vector. It has magnitude – just as speed does – but it also has a direction. When we talk about speed, we don’t care what about the direction of motion. The car went at a speed of 50 miles per hour. We don’t care if it went south, north, east, west, whatever. With velocity we do care about the direction. Velocity would be the motion of a car that is going south at 35 mph.

Because we care about speed and direction with velocity, an object can be moving at a constant speed, but if it is changing direction at the same time, it’s velocity is changing too. For example, a toy train traveling around a circular track is moving at a constant speed. It does not have a constant velocity, however, because its direction is constantly changing.

There are three types of speed and three types of velocity:

Instantaneous speed / velocity: The speed or velocity at an instant in time. You look down at your speedometer and it says 20 mph. You are traveling at 20 mph at that instant. Your speed or velocity could be changing, but at that moment it is 20 mph.

Average speed / velocity: If you take a trip you might go slow part of the way and fast at other times. If you take the total distance traveled divided by the time traveled you get the average speed over the whole trip. If you looked at your speedometer from time to time you would have recorded a variety of instantaneous speeds. You could go 0 mph in a gas station, or at a light. You could go 70 mph on the highway, and only go 30 mph on surface streets. But, while there are many instantaneous speeds there is only one average speed for the whole trip.

Constant speed / velocity: If you have cruise control you might travel the whole time at one constant speed. If this is the case then your average speed will equal this constant speed.

Constant velocity must have both constant magnitude and constant direction.

The very first Physics Equation

Here you are, your first Physics Equation…

The Constant Velocity Equation

[pic]

Where v is velocity, ∆x is displacement, and ∆t is the time interval

We use this equation whenever an object’s velocity is constant. That means both the speed and direction must be constant in order for this equation to be valid. If either of those quantities is changing, we do not have constant velocity and must use different equations to describe the motion.

We can also use this equation whenever we want to know the average velocity.

• Example 3: In the 1988 Summer Olympics, Florence Griffith-Joyner won the 100 m race in a time of 10.54 s. What was her average velocity?

[pic]

[pic] [pic] [pic]

• Example 4: A car travels for 60 minutes at a constant velocity of 25 m/s. How far does the thing go?

[pic] [pic] [pic]

All Motion Is Relative:

Motion, i.e. velocity, is said to be relative. This is an important concept. What it means is that when we say that something has a given velocity, that velocity is relative to something else (these are called reference frames). So a car traveling to the east at 125 km/h is doing so relative to the earth. As you sit in whatever room you are currently in you are not moving (you have no motion) with respect to the room. However, the room and everything in it is rotating around the center of the earth. Not only that, but the earth itself is moving around the sun in its orbit! The solar system is moving around the center of the galaxy! The galaxy (and everything in it) is also moving away from the center of the universe!

If you are a passenger in an aircraft traveling at 500 mph over the earth, you are moving at 500 mph relative to the earth, but have no motion relative to the plane, unless you get up and go walking in the aisle, then you might have a motion relative to the plane, of, say, 3 mph. Depending on which way you go, your motion relative to the earth could be 503 mph or 497 mph.

4. Use the information provided in this packet to answer the following problems and questions. For each problem, list the original equation used, show correct substitution, and arrive at the correct answer with the correct units. All velocities must be reported in m/s, displacements in meters, and time in seconds.

a. You begin a trip and record the odometer reading. It says 45545.8 miles. You drive for 35 minutes. At the end of that time the odometer reads 45569.8 miles. What was your average speed in miles per hour?

b. A high speed train travels from Paris to Lyons at an average speed of 227 km/h. If the trip takes 2.0 h, how far is it between the two cities?

c. Give an example of two cars that have the same speed but different velocities.

d. You are driving down the road at a constant velocity. What are 3 ways you could safely change your velocity?

e. You nose out another runner to win the 100.0 m dash. If your total time for the race was 11.80 s and you aced out the other runner by 0.001 s, by how many meters did you win?

f. The speed of sound is 344 m/s. You see a flash of lightning and then hear the thunder 1.5 seconds later. How far away from the lightning strike are you?

g. A train travels from Denver to Bougainvillea in 5 hours and 37 minutes. If the average speed for the train was 76.5 km/h, how much distance did it cover?

Section IV: Constant Acceleration

With constant velocity, an object is moving with a constant speed and in a constant direction. No turns, no slowing down, no speeding up. So what if speed or direction is changing? What do we call that? Well, if an object’s speed or direction is changing we have changing velocity. We call that acceleration.

Acceleration: The rate at which an object’s velocity changes. Acceleration is given the symbol a. Acceleration is a vector quantity, just like velocity.

[pic]

• Example 5: A plane goes from rest to speed of 235 km/h in 15.0 s. Find the acceleration.

[pic]

This answer, 4.35 m/s2, means that the velocity changes by 4.35 m/s every second. At the end of the first second it is 4.35 m/s, after two seconds it is 8.70 m/s, after three seconds it is 13.0 m/s, after four seconds it would be 17.4 m/s, etc. The unit for acceleration is m/s2. When we see this, we say “meters per second squared.” We will always use m/s2 for the units for acceleration. This is because things on earth don’t accelerate for long periods of time. Very few things can actually accelerate for more than a few seconds.

• Example 6: A car slows from 85.5 m/s to a speed of 33.2 m/s in 1.25 s. Find the acceleration.

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The minus sign means that the acceleration is in the opposite direction from the velocity. It only means that the car is moving in one direction, but the acceleration points in the opposite direction. You do this every time you approach a stop sign while driving. Please keep in mind, however, that a negative acceleration does NOT mean that an object is necessarily slowing down. An object can be speeding up and have a negative acceleration if the velocity and acceleration are in the opposite direction.

So, this section is titled “Constant Acceleration.” But, what does constant acceleration mean? If an object is undergoing a constant acceleration, its velocity is changing at a constant rate. Every second that it undergoes this constant acceleration, its velocity changes. When an object is undergoing constant acceleration, we can analyze the motion and come up with three equations that will describe the motion. We will call these equations the constant acceleration equations. Here’s how we derive the first one:

Start with the equation for acceleration: [pic] [pic] [pic]

This is the first of our constant acceleration equations. There are three equations that we will be concerned with that will be introduced here in this summer assignment. Here they are:

[pic] [pic] [pic]

These are collectively known as the constant acceleration equations or kinematics equations. Their purpose is to explain quantitatively the motion of an object undergoing constant acceleration. I will present for you three examples using each of them and then ask that you solve a few problems using them. For each of the equations, the variables are defined this way:

|vf = final velocity |vi = initial velocity |Δt = time |a = acceleration |

| | | | |

|xf = final position |xi = initial position | | |

• Example 7: A cheetah is hiding in some grass, waiting for a hapless gazelle to wander by so she can have some lunch. When one does, she bolts out of her hiding place and reaches her top speed of 27 m/s (60 mph) in 3.0 seconds. What was her rate of acceleration?

[pic]

• Example 8: A car is waiting for its turn to go at a stop light. When the light turns green, it takes off. The car’s rate of acceleration is 4.0 m/s2. How far has the car gone after 5.0 seconds?

• Example 9: A bicyclist is at rest at the top of a 20 meter long hill. When he starts to go, he accelerates at a constant 3.0 m/s2. How fast is he going at the bottom of the hill?

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5. Use the three kinematics equations to solve each of the problems below. Read each problem, think about what is given and what is not given, and decide which equation you can eliminate to come to a solution.

a. A car traveling at 7.0 m/s accelerates at 2.5 m/s2 to reach a speed of 12.0 m/s. How long does it take for this acceleration to occur?

b. A person pushes a stroller from rest and accelerates at a rate of 0.50 m/s2. What is the velocity of the stroller after it has traveled 4.75 m?

c. An airplane starting at rest at one end of a runway undergoes a uniform acceleration of 4.8 m/s2 for 15.0 s before takeoff. How long must the runway be for the plane to be able to take off?

d. A bus slows down with an acceleration of -1.8 m/s2. How long does it take the bus to slow from 9.0 m/s to a complete stop?

e. A car with an initial speed of 4.3 m/s accelerates uniformly at 3.0 m/s2 for 5.0 s. How far has the thing gone?

f. A train slows down with a constant acceleration from an initial velocity of 21 m/s to 0 m/s in 21.0 s. How far does it travel before stopping?

g. A sailboat starts from rest and accelerates constantly for 52 seconds over a distance of 280 m. What is the boat’s velocity at this point?

Those were all fairly simple. This next set of problems poses a bit more complication. Remember, we will always be working with kilograms, meters, and seconds as our units, so other units must be converted.

6. Use the three kinematics equations to solve each of the problems below. Read each problem, think about what is given and what is not given, and decide which equation you can eliminate to come to a solution.

a. A racecar accelerates from rest to a speed of 287 km/h in 6.8 seconds. What is its average acceleration?

b. When striking, the pike, a predatory fish, can accelerate from rest to a speed of 4.0 m/s in 0.11 s. (a) What is the acceleration of the pike during this strike? (b) How far does the pike move during this strike?

c. A driver has a reaction time of 0.50 s, and the maximum deceleration of her car is 6.0 m/s2. She is driving at 20 m/s when suddenly she sees an obstacle in the road 50 m in front of her. Can she stop the car in time to avoid a collision?

d. A light rail train going from one station to the next on a straight section of track accelerates from rest at 1.4 m/s2 for 15 s. It then proceeds at constant speed for 1100 m before slowing down at 2.2 m/s2 until it stops at the station. (a) What is the distance between the stations? (b) How much time does it take the train to go between stations?

e. A speed skater moving across frictionless ice at a constant 8.0 m/s hits a 5.0 m wide patch of rough ice. She slows steadily, then continues on at a constant 6.0 m/s. (a) Sketch a velocity-time graph of this motion (it doesn't need to be exact, just get the basic shape of the graph) (b) What is her acceleration while on the rough ice?

f. You are designing an airport for small planes. One kind of airplane that might use this airfield must reach a speed before takeoff of at least 27.8 m/s, and can accelerate at 2.0 m/s2. (a) If the runway is 150 m long, can this airplane reach the required speed for take off? (b) If not, what minimum length must the runway have?

Section IV: Visualizing Motion

Motion is taking place all around us at all times. Microscopically, atoms are in constant motion. The stars and planets are also in constant motion. Throughout any given day you move by walking, running, driving, or in a variety of other ways.

Motion is a theme that will appear in one form or another throughout this entire course. Although we all have ideas and intuition about motion based on our experiences, some of the important aspects of motion turn out to be rather subtle. So, in order to get a conceptual grasp of motion, this section of the summer assignment will help you to visualize some of these more subtle aspects of motion. If you have had some difficulty with the problems you’ve encountered so far in this assignment, hopefully this section will give you some tools you can use to help you.

One way to visualize motion is through using a motion diagram. In a motion diagram, an object is represented as a point mass. What that means is we represent the mass of the entire object as a single point. To do this, we represent the object as a dot. We then space out successive dots in such a way as to characterize the object’s velocity. For example, if the object is moving at a constant velocity, we place dots at equal intervals to show that the object covers equal distance intervals in equal time intervals. If the object is accelerating, we have to show the dots getting further and further apart or closer and closer together.

Motion diagrams are very useful in understanding what is happening in a problem and what needs to be done to solve a problem. The critical component of a motion diagram is that the amount of time between each dot is the same. Complete motion diagrams contain the following elements:

1. Dots indicate object's position at successive times

2. Time interval between dots is the same (1 s, 5 min, etc.)

3. Arrows connecting the dots show velocity vectors

4. Longer arrows means greater velocity

5. The direction of any acceleration is toward the longest velocity arrows

Example #10: Draw a motion diagram of the following situations…

a) A girl walking at a constant speed for 10 seconds.

b) A bicyclist starting from rest, speeding up for 4 seconds, and then going at a constant speed for 6 seconds.

c) A car starting from rest, accelerating for 3 seconds, going at a constant speed for 4 seconds, and then slowing and stopping for 3 seconds.

7. To practice motion diagrams, draw one for each of the problems below. These are problems you’ve seen already, so don’t worry about solving them, just draw a motion diagram for them.

a) A racecar accelerates from rest to a speed of 287 km/h in 6.8 seconds. What is its average acceleration?

b) A train slows down with a constant acceleration from an initial velocity of 21 m/s to 0 m/s in 21.0 s. How far does it travel before stopping?

c) A light rail train going from one station to the next on a straight section of track accelerates from rest at 1.4 m/s2 for 15 s. It then proceeds at constant speed for 1100 m before slowing down at 2.2 m/s2 until it stops at the station. (a) What is the distance between the stations? (b) How much time does it take the train to go between stations?

Section V: Free-Fall

Aristotle (384 - 322 BC) said that things fall because they want to regain their natural state - earth with earth, water with water, and so on. Thus a rock will fall back to the earth to be with the other rocks. Since a big rock possesses more "earth", it will fall faster than would, say, a feather. Aristotle’s idea appears to be true because a rock certainly falls faster than a feather. In fact it made so much sense, that Aristotle's ideas on the subject were the accepted truth for around 2000 years until the Renaissance.

The first scientific study of gravity was done by Galileo Galilee (1564 - 1642). He was trained as a mathematician and was a university professor. In the late 1500's Galileo conducted a series of experiments on gravity. He is supposed to have demonstrated that heavy objects and light objects fall at the same speed. The act of doing experiments to find out what would happen was a very daring idea.

Galileo did not, as is popularly believed, state that the objects would hit the ground at the same time - he understood air resistance. He also understood that without air resistance, the objects would fall at exactly the same rate.

You can actually try this yourself. Find two objects of different masses and negligible air resistance, drop them from the same height, and you will see that they hit the ground at the same time. Galileo's idea that things fall at the same rate flies in the face of common sense. It seems reasonable that heavy things ought to fall faster than light ones, but that just is not the case.

To study gravity, Galileo found that he had to slow it down. This was because he couldn’t measure the time it took an object to fall with the crude instruments of the time. Gravity was “slowed down” by having balls roll down inclined planes (ramps). Gravity still caused the motion, but its effect was decreased to the point where Galileo could gather useful data. Galileo found that the distance that accelerated objects would travel was proportional to the square of the time.

Acceleration due to Gravity: On the earth, gravity exerts a force on everything with mass. (A force is a push or pull.) This force makes all objects accelerate downwards, towards the center of the earth. This acceleration varies a tiny little bit depending on where you are - at the North Pole this acceleration is 9.83217 m/s2 and at the Equator it has a value of 9.78039 m/s2. This is because the earth is not a perfect sphere. Fortunately we can safely ignore the tiny differences in the acceleration of gravity. The value which is commonly used for this acceleration is 9.80 m/s2. In English units it is 32.0 ft/s2. Gravity's acceleration is kind of special so it is given its very own little symbol, g.

[pic]

Drop a rock from the top of a cliff and, in one second, it will reach a speed of 9.80 m/s, after two seconds it will be traveling at 19.6 m/s, in three seconds it’s going 29.4 m/s, at four seconds its speed will be up to 39.2 m/s, and so on. It looks like the rock will keep going faster and faster and faster until it smashes into the earth, and it would, if it were falling in a vacuum. The thing is, see, that the air causes a frictional force that opposes the rock's fall and slows it down. For short drops with dense objects (like rocks) we can reasonably ignore the effects of the air. Oh, the fancy, scientific term for this force exerted by the air is drag or air resistance, sometimes it is called wind resistance. At high velocities or over long distances, the drag can become significant, especially for objects that are not dense, like feathers or leaves or fluff. In the real world, an object in free fall will accelerate to its terminal velocity. This is the speed at which the force of gravity equals the drag force. The object then stops accelerating and falls at a constant velocity. People jumping out of airplanes experience this. The typical laid out position that sky divers use gives them a terminal velocity of around 100 mph.

When an object is released and allowed to fall, its motion can be described by the following table (ignoring air resistance):

The constant acceleration equations can be used to describe the motion of falling objects. However, instead of using x as our position we use y and for acceleration we use g instead of a. Therefore, the constant acceleration equations become...

vf = vi – gΔt

yf = yi + viΔt – ½ gΔt2

vf2 = vi2 – gΔy

Negative or Positive: By convention, down is usually considered to be the negative direction. Because free fall acceleration always points downward, it is substituted into the equations as a negative. Does this mean the object is decelerating (slowing) or does it mean that the object is moving along a negative (perhaps the y) axis? For an object falling along the y-axis, due to gravity, it means the object is accelerating, but in the downward direction.

To make sense of free-fall, let’s look at a motion diagram of a ball being tossed straight up and then falling back down. Try it first, toss a ball straight up and catch it. That’s the motion we’re looking at. Here is a motion diagram of such a thing…

From this diagram, you can see that as the ball rises to its highest point, it slows and the acceleration points downward. As the ball comes back down, it speeds up and the acceleration points downward as well. For an object in free fall, the acceleration always points downward with a magnitude of 9.80 m/s2. Here are other important points summarizing this type of motion…

• Velocity is first upward, then downward

• Acceleration always points downward.

• The distance up is equal to the distance down

• The launch speed is equal to the speed just before landing

• The speed at the highest point is 0.0 m/s

• The time up equals the time down

• Example #11 - A ball is thrown straight upward. (a) If it takes 2.25 seconds to reach the top of its path, what is its initial speed? (b) What total distance does the ball go? (up and back down) (c) What is the ball’s displacement?

a) Since the ball is traveling upward, and the acceleration is downward, the ball will slow down as it moves up. For the upward part of its motion, its final velocity will be zero - it will then momentarily come to rest and then change direction and begin to accelerate downward. Since we know that for the upward part of its journey the final velocity is zero, we can easily calculate the initial velocity.

b) There are multiple ways to solve this part of the problem. I’ll show you two of them. First, since you know it takes 2.25 s to reach the highest point, and you can figure out the height it goes using the third equation, substituting in the initial velocity from part (a), and doubling the answer. The second way would be to use the time up (2.25s) and the initial velocity calculated from part (a) in the second equation, and again doubling the answer.

c) Because the ball goes up and back down to the same point, it is not displaced from its original position. Therefore, its displacement is zero. You can see this by using the second equation and the total time (4.5 s).

• Example #12 - A stone is launched straight up from the top of a large building with an initial speed of 7.5 m/s. (a) How high does it go from the top of the building? (b) How much time to reach the maximum height? (c) If the building is 45.2 m tall, how much time will it take to hit the ground from when it was initially launched?

(a) To find how high it goes, simply use the third equation…

(b) For this part, we want to find out how much time it took to get to the highest point. Based on what we are given, the most logical way to solve the problem will be to use the first equation…

(c) For this part, we want to find out how much total time it takes the ball to rise above the top of the building and then fall to the ground below. There are a couple of ways to answer this, but I’m going to show you only one of them. We know it takes 0.77 s to reach the highest point, that at that point its velocity will be 0.0 m/s, and that the highest point is 2.87 m above the top of the building. So, if we look from the perspective of the highest point, we can use the second equation with a total distance of 48.07 m and solve for the time as if the ball had been dropped from that point. Then, we simply add the time from part (b) to get the final answer. Here’s how it would look…

8. Use the three kinematics equations to solve each of the problems below. Read each problem, think about what is given and what is not given, and decide which equation you can eliminate to come to a solution.

a) Josh’s truck falls straight down off a cliff. If the cliff is 33.5 m high, how much time for the truck to reach the bottom?

b) Andrea tosses a ball straight up in the air so that it goes up, comes down, and she catches it. If it took 5.6 s from when she threw it to when she caught it, how high did it go?

c) In 1947 Bob Feller, a pitcher for the Cleveland Indians, threw a baseball across the plate at 98.6 mph or 44.1 m/s. For many years this was the fastest pitch ever measured. If Bob had thrown the pitch straight up, how high would it have gone?

d) Samantha is on top of a building that is 75.0 m tall. She launches a ball out of a mechanical launcher straight up with an initial velocity of 33.8 m/s. (a) How high does the ball travel? (b) It goes up and then falls down to the ground below. How much total time is it in the air?

e) Jason throws a ball vertically upward with a speed of 19.6 m/s. (a) What are the ball's velocity and height after 1.00, 2.00, 3.00, and 4.00 s? (b) Draw the ball's velocity vs. time graph. Give both axes an appropriate numerical scale.

f) In an action movie, the villain is rescued from the ocean by grabbing onto the ladder hanging from a helicopter. He is so intent on gripping the ladder that he lets go of his briefcase of counterfeit money when he is 130 m above the water. If the briefcase hits the water 6.0 s later, what was the speed at which the helicopter was ascending?

Section VI: Final Thoughts

Congratulations! You have completed the summer assignment. Below are select answers to some of the problems you encountered. Check your answers, find any mistakes, correct them, and use that information to help you get a sense of how well you’ve done on problems not included below. If you have questions, please email me at aguaty@mdc.edu and I will get back to you.

|#1 |#2 |#3 |#4 |#5 |#6 |#7 |#8 | |(a) |[pic]

|i. Tangent

iii. 0.35 m | |vavg = 41.2 mi/hr | |a = 11.72 m/s2 | | | |(b) |[pic]

[pic][pic] |

23o |b = 15 m/s

c = 21.2 m/s | | | |See Below |Δy = 38.4 m | |(c) |

| | | |xf = 540 m |xtotal = 43.3 m

She stops in time | | | |(d) |

|

40o | | | |Δxtotal = 1357.7 m

Δttotal = 76.9 s | |Δy = 58.3 m

Δttotal = 8.67 s | |(e) |[pic]

| | |Δx = 0.0085 m |xf = 59 m | | | | |(f) |

| | |Δx = 516 m | |vf = 24.5 m/s (can’t make it)

Δx = 193.21 m | |vi = 7.73 m/s | |(g) |

| | | |vf = 10.8 m/s | | | | |(h) |[pic]

| | | | | | | | |

#7b

-----------------------

(

30o

C

67o

B

A

Figure C

Figure B

Figure A

x

y

40o

θ

A

4

20

12

B

[pic]

Table of Standard Prefixes

Prefix Symbol Factor

giga G 109

mega M 106

kilo k 103

----------------------------------------------------------------------ⴭⴭⴭⴭഭ散瑮३उ按उउ〱㈭洍汩楬उउ७उㄉⴰള業牣९उ⠉उउ〱㘭渍湡९उ渉उउ〱㤭瀍捩९उ瀉उउ〱ㄭല഍吨楨⁳獩愠潢瑵㔠‶業敬⥳഍桔獩瀠潲汢浥琠汥獬甠⁳桴瑡琠敨挠敨瑥桡猠慴瑲⁳畯⁴楨楤杮椠潳敭朠慲獳‮潓眠⁥湫睯琠慨⁴敨⁲湩瑩慩敶潬楣祴洠獵⁴敢稠牥⹯ഠ瑉琠汥獬甠⁳桴瑡猠敨爠慥档獥愠琠灯猠数摥漠⁦㜲洠猯‮潓眠⁥湫睯琠慨⁴敨⁲楦慮敶潬楣祴洠獵⁴敢㈠‷⽭⹳ഠ瑉琠汥獬甠⁳桴瑡猠敨爠慥档獥琠楨⁳潴⁰灳敥⁤湩㌠〮猠捥湯獤‮潓琠浩⁥畭瑳戠⁥⸳‰⹳ഠ敗愠敲渠瑯琠汯⁤湡瑹楨---------

centi c 10-2

milli m 10-3

micro ( 10-6

nano n 10-9

pico p 10-12

(This is about 56 miles)

• This problem tells us that the cheetah starts out hiding in some grass. So we know that her initial velocity must be zero.

• It tells us that she reaches a top speed of 27 m/s. So we know that her final velocity must be 27 m/s.

• It tells us that she reaches this top speed in 3.0 seconds. So time must be 3.0 s.

• We are not told anything about her position, so we don’t know anything about xi or xf. This eliminates any ability to use the second or third equation, meaning we must use the first equation to solve.

• This problem tells us that the car starts out at a stop light. So vi = 0.0 m/s.

• It tells us that the car has an acceleration of 4.0 m/s2. So a = 4.0 m/s2.

• It tells us that we are interested only in the first 5.0 seconds of the car’s motion. So Δ t = 5.0 s.

• Because we are starting at a stop light, we consider that point to be the initial position. So xi = 0.0 m.

• We are asked to find how far the car goes. In other words, we are asked to find the final position of the car after the 5.0 seconds. We are not told anything about the car’s final velocity, so that eliminates being able to use the first or third equations. So we must use the second equation to solve.

[pic]

• This problem tells us that the bicyclist starts out at rest at the top of the hill. So vi = 0.0 m/s and xi = 0.0 m

• It tells us that a = 3.0 m/s2.

• It tells us that the hill is 20.0 meters long. So xf = 20.0 m.

• We are not told anything about the amount of time it takes to get to the bottom of the hill so the first and second equations can be eliminated. We use the third equation to solve.

0 s

10 s

a = 0.0 m/s2

In this motion diagram, all the dots are spaced equally because the girl is walking at a constant speed. The arrows are all the same length, indicating that her velocity does not change. Because the arrows are all the same length, you know that her acceleration is zero.

0 s

10 s

4 s

a

a = 0.0 m/s2

In this motion diagram, the dots get further and further apart for the first 4 seconds and then are equally spaced for the last 6 seconds. The arrows get longer and longer at first, showing that the velocity increases. The direction of acceleration during the first 4 seconds is in the direction of the longest arrows, as shown. This would be a positive acceleration. There is no acceleration for the last 6 seconds.

0 s

10 s

3 s

a

a = 0.0 m/s2

7 s

a

In this motion diagram, the dots get further and further apart for the first 3 seconds, are equally spaced for the next 4 seconds, and get closer and closer together for the last 3 seconds. The length of the arrows indicates the object is speeding up, going at a constant velocity, and then slows to a stop. Notice that the acceleration points in the positive direction for the first 3 seconds, but in the negative direction for the last 3 seconds.

0 s

2.5 s

1.25 s

a = g

a = g

0 s

4.5 s

2.25 s

a = g

a = g

[pic]

Second Way

First Way

[pic]

[pic]

[pic]

This time our object is thrown upward, but when it comes back down it goes a significant distance further than it did going up. That is, it goes up and comes back down farther. Let’s look at each part of this motion…

y = ?

y = 45.2 m

vi = 7.5 m/s

v = 0.0 m/s

[pic]

[pic]

[pic]

There are a couple of other ways to solve this part of the problem. Challenge yourself by seeing if you can figure out what they are

0 s

3 s

a

6 s

9 s

12 s

15 s

18 s

21 s

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