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Mathematical Physics

Constant Speed Motion

2019-02-06 www.njctl.org

· Motion in One Dimension · Constant Speed · Distance vs Time Graphs · Average Speed · Position and Displacement · Velocity · Position-Time Graphs · Formulas and Solving Problems

Table of Contents: Constant Speed Motion

Click on the topic to go to that section. Move any photo or image in this presentation to reveal a link to its source, providing attribution and additional information. · Bowling Ball Lab

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Motion in One Dimension

How can you determine how fast a toy car is moving?

· If you had a pull-back toy car, what would you need to find its speed? Speed Quick-Lab In your lab notebook... · Write your approach to determining the speed · Record any measurements and calculations of speed · Carefully note everything you notice about the cars motion from start to finish

How can you determine how fast a toy car is moving?

Discussion: Conclusions from Quick Lab · · · · · ·

Many of you found the speed by...

· Identifying a distance for your car to travel · Finding the time it took your car to travel that distance · But what exactly are distance and time?

Distance

We all know what the distance between two objects is...

• r do we?

Can you define it without using the words length or distance?

Distance

Here's what Wikipedia says: "Distance ... is a numerical description of how far apart objects are. In physics or everyday usage, distance may refer to a physical length..." Does this really help? Distance is a fundamental property. It is so fundamental that the words we use to define it are just synonyms: farness, length... To know what a distance is we can only compare it to another distance.

Distance

We can compare the distance between two objects to the distance between two other objects. For convenience, we create standard distances so that we can easily make comparisons... and tell someone else about them.

Distance

There are many units that people have used for distance and length... including inches, feet, hands, rods, chains, furlongs, miles, ... BUT The meter is the world's standard unit for measuring distance. This doesn't really define distance, but it allows us to work with it. d = 0.2 m

Distance - Symbol

We will use symbols as a shorthand for all physics quantities. It will make it easy for us to take notes, write down relationships (formulas) and work through questions. More later. The symbol for distance is "d". The unit we'll use for distance is the meter, abbreviated m.

Time

Just as with distance, everyone knows what time is... Try defining time... without using the word time.

Time

Here is what Wikipedia says: "Time is ... a measure in which events can be ordered from the past through the present into the future, and also the measure of durations of events and the intervals between them. Time has long been a major subject of study in religion, philosophy, and science, but defining it in a manner applicable to all fields without circularity has consistently eluded scholars." It looks like we are a bit stuck here too!

Time

Like distance, time is a fundamental aspect of nature. It is so fundamental that it is impossible to define. Everyone knows what time is, but no one can really say what it is... But as with distances, times can be compared.

Time

We can say that in the time it took to run 100 meters, the second hand of my watch moved 10 "tick marks" or seconds. When we compare the time between two events to the time between two

• ther events, we are measuring time.

This doesn't define time, but it does allow us to work with it.

t = 10 s

Time - Symbol

Here is the next shorthand symbol for a physical quantity. The symbol for time is "t". The unit we'll use for time is the second, abbreviated s.

Speed

In our toy car lab, we established the definition of speed... "speed is the distance traveled divided by the time it took to travel that distance." Or using the symbol "s" to mean speed...

Speed

The unit of speed can be found by substituting the units for distance and time into the equation we read this as "meters per second"

Speed

miles hour Other commonly used units for speed are: kilometers hour feet second What do these have in common? Can you make up another unit for speed? "miles per hour" Or "mph" "kilometers per hour" Or "kph" "feet per second" Or "fps"

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Formulas and Solving Problems

Formulas

A formula is a mathematical statement that describes a relationship between two or more quantities (or variables) - it always stays the same. In words, we could say that "speed is the ratio of distance traveled divided by the time it takes to travel that distance". But the mathematical formula above is more concise. The letters are placeholders for the actual numbers that you will find or need to calculate in each problem. Finally, formulas can be manipulated, with the rules of mathematics, to create new formulas. is our first physics formula.

Formulas

In general, if your goal is to solve a formula for a numerical answer, you need one formula, or equation, for each unknown variable. One formula allows you to solve for one unknown.

Solving Word Problems

In physics (and in many other disciplines), you will need to solve a problem based on information you read or are told. Here is the approach for the simplest problems: 1. Highlight the given quantities and the quantity to find, then list them. 2. Draw and label a diagram of the situation described. 3. Select the formula only containing the quantities given and to find. 4. Manipulate the formula so that the quantity to find is on the left (you'll learn how to solve for the quantity to find shortly). 5. Substitute for the given quantities. 6. Calculate the quantity you need to find. Box the answer with its units.

Solving Word Problems

Use this template when solving a problem. The next slide has a template you can print out. Let's try an example! Given Find Diagram Formula Manipulate Substitute Answer

Given Find Formula Manipulate Substitute Answer Diagram

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate Substitute Answer

• 1. highlight the given quantities and

the quantity to find, then list them.

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate Substitute Answer

• 2. Draw and label a diagram of

the situation described.

d = 50 m t = 20 s

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate Substitute Answer

• 3. Select the formula only containing

the quantities given and to find.

d = 50 m t = 20 s

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate s on left, so no manipulation needed Substitute Answer

• 4. Manipulate the formula so that

the quantity to find is on the left.

d = 50 m t = 20 s

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate s on left, so no manipulation needed Substitute Answer

• 5. Substitute for the given

quantities.

d = 50 m t = 20 s

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate s on left, so no manipulation needed Substitute Answer

• 6. Calculate the quantity you need to
• find. Box the answer with its units.

d = 50 m t = 20 s

Example: Solving for Speed

A car travels 50m in 20s; What is its speed? Given d = 50 m t = 20 s Find s = ? Diagram Formula Manipulate s on left, so no manipulation needed Substitute Answer

d = 50 m t = 20 s

CHECK: are the units in the answer (m/s) the units for speed?

Solving Word Problems

You may be thinking... why go through all this for such a simple problem? I could do this in my head or just scribble 50/20 and stick it my calculator. But there is a reason for practicing this problem solving approach with such easy problems... Soon problems will be more complicated. If you learn to always use this approach, you will be able to keep your work straight and be much more likely to get the right answer! Use the method in the previous example to solve the next two word problems.

1 An airplane on a runway moves 500 m in 10 s; what is the airplane's speed?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

2 A rabbit hopped a distance of 60 meters in 20 seconds; what was the speed of the rabbit?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

Solving for Distance

So far, we have found the speed of an object if we are given: · how far it traveled and · how long it took to travel that far Sometimes we want to know how far something has traveled if we know: · its speed and · the amount of time it has traveled Any ideas of how we could figure out an equation to do that?

Our goal is to be able to solve any equation for any variable that appears in it. Let's look at a our formula first. The variables in this equation are s, d and t. Solving for a variable means having it alone on the left side. This equation is currently solved for "s".

Solving for a Variable

Solving for a Variable

Remember! · The letters are placeholders for the actual numbers that you will find or need to calculate in each problem. · Formulas can be manipulated, with the rules of mathematics, to create new formulas.

Like in any game there are a few rules.

• 1. To "undo" a mathematical operation, you must do the opposite.
• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the

• ther.
• 3. If there is more than one operation going on, you must undo

them in the opposite order in which you would do them, the

• pposite of the "order of operations."
• 4. You can always switch the left and right sides of an equation.

The Rules for Solving for a Variable

Let's solve this equation for "d"

That means that when we're done we'll have d alone on the left side

• f the equation.

The Rules

3 First, is d already alone ? If not, what is with it? A s B d C t D it is already alone

Answer

4 What mathematical operation connects d and t? A d is added to t B d is multiplied by t C d is divided by t D t is subtracted from d

Answer

5 What is the opposite of dividing d by t? A dividing t by d B dividing by v into t C multiplying d by t D multiplying by t by d

Rule 1. To "undo" a mathematical operation, you must do the opposite. Answer

6 What must we also do if we multiply the right side by t? A divide the left side by t B multiply the left side by t C divide the left side by d D divide the left side by d

Rule 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. Answer

7 Is there more than one mathematical operation acting on "d"? Yes No

Rule 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite

• f the "order of operations."

Answer

The order of s and t doesn't matter. Are we done?

Applying Rules 1 and 2

• 1. To undo a mathematical operation, you must do the opposite.
• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the other. So we undo d being divided by t, by multiplying both sides by t.

We've now solved our equation for d. A harder problem will to solve it for t. Before we go on to solve for t, lets use our problem solving approach for problems where we need to find d.

Applying Rule 4

Rule 4. You can always switch the left and right sides of an equation.

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? What should we do first?

Given Find Diagram Formula Manipulate Substitute Answer

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel?

Given s = 3 m/s t = 2.5 s Find d = ? Diagram Formula Manipulate Substitute Answer

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel?

Given s = 3 m/s t = 2.5 s Find d = ? Diagram

t =0 t=2.5 s s = 3 m/s s = 3 m/s

Formula Manipulate Substitute Answer

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel?

Given s = 3 m/s t = 2.5 s Find d = ? Diagram

t =0 t=2.5 s s = 3 m/s s = 3 m/s

Formula Manipulate Substitute Answer

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel?

Given s = 3 m/s t = 2.5 s Find d = ? Formula Manipulate Substitute Answer

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram

t =0 t=2.5 s s = 3 m/s s = 3 m/s

Given s = 3 m/s t = 2.5 s Find d = ? Formula Manipulate

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram

t =0 t=2.5 s s = 3 m/s s = 3 m/s

Substitute Answer

Given s = 3 m/s t = 2.5 s Find d = ? Formula Manipulate

Solving for Distance

A ball rolled at a speed of 3 m/s for 2.5 s; how far did it travel? Diagram

t =0 t=2.5 s s = 3 m/s s = 3 m/s

Substitute Answer d = 7.5 m

8 A car was driven at a speed of 40 m/s for 4 s; what distance did the car travel?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

9 How far will you go if you run at a speed of 20 m/s for 6 s?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

Finding Time

So far, we have found the speed of an object if we are given how far it traveled and how long it took to travel that far. And, we have found how far an object has traveled if we know its speed and travel time. But, what if we want to know how much time it will take for an

• bject to travel a given distance at a given speed?

How would we could figure out an equation to do that?

Let's solve for "t" That means that when we're done, t will be on its own on the left side of the equation.

Solving for Time

10 Is t already alone? If not, what is with it? A s B d C t D it is already alone

Answer

11 What mathematical operation connects d to t? A t is being divided by d B d is being divided by t C d is being multiplied by t D t is being subtracted from d

Answer

12 What is the opposite of dividing d by t? A dividing d by t B dividing s by t C multiplying d by t D multiplying t by d

Rule 1. To "undo" a mathematical operation, you must do the opposite. Answer

13 What must we do if we multiply the right side by t? A divide the left side by t B multiply the left side by t C divide the left side by d D divide the left side by d

Rule 2. You can do anything you want (except divide by zero) to one side of an equation, as long as you do the same thing to the other. Answer

14 Is there more than one mathematical operation acting on

"d"? Yes No

Rule 3. If there is more than one operation going on, you must undo them in the opposite order in which you would do them, the opposite

• f the "order of operations."

Answer

Solving for t

• 1. To "undo" a mathematical operation, you must do the opposite.
• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the other. So we undo d being divided by t, by multiplying both sides by t.

• r

Are we done?

15 Is t on its own? If not, what is it with? A s B d C t D it is already alone

Answer

16 What mathematical operation connects s to t? A t is being divided by d B t is being divided into s C t is being multiplied by s D t is being subtracted from s

Answer

17 What is the opposite of multiplying t by s? A dividing t by s B dividing t by t C multiplying t by t D multiplying t by s

Answer

• 1. To "undo" a mathematical operation, you must do the opposite.

Solving for t

• 2. You can do anything you want (except divide by zero) to one

side of an equation, as long as you do the same thing to the

• ther.

18 Is t alone on the left? If not, what is it with? A s B d C t D it is alone

Answer

Solving for Time

A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m?

Given Find Formula Manipulate Substitute Answer

Solving for Time

A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m? Diagram

Given s = 2 m/s d = 9 m Find t = ? Formula Manipulate Substitute Answer

Solving for Time

A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m? Diagram

s = 2 m/s s = 2 m/s d = 9 m

Given s = 2 m/s d = 9 m Find t = ? Formula Manipulate Substitute Answer t = 4.5 s

Solving for Time

A ball rolls at a speed of 2 m/s; How much time will it take the ball to travel 9 m? Diagram

s = 2 m/s s = 2 m/s d = 9 m

19 How much time will it take you to travel a distance of 120 m if you can move at a constant speed of 20 m/s?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

20 How much time will it take you to travel a distance of 150 m at a speed of 30 m/s?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

Why do Algebra to Rearrange the Formula?

Why not just plug numbers into the original formula? Suppose you wanted to know how far you can travel at different speeds for different times. speed [mph] time [hours] distance [miles] 30 1 30 2 45 3 60 2 Would you rather... · plug the numbers into and solve for d 4 times? -OR- · plug numbers 4 times into the new formula you derive once?

Why do Algebra to Rearrange the Formula?

Why not just plug numbers into the original formula? Suppose you wanted to know how much time it takes you to travel at different distances at different speeds? distance [miles] speed [mph] time [hours] 200 40 200 50 300 50 300 60 Would you rather... · plug the numbers into and solve for d 4 times? -OR- · plug numbers 4 times into the new formula you derive once?

21 A truck was driven at a speed of 35 m/s for 12 s; what distance did the car travel?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

22 An boat sails 800 m in 120 s; how fast is the boat moving?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

23 How much time will it take a airplane to travel 2000 m if it can maintain a constant speed of 160 m/s?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

24 How far will a tortoise go in 240 s if it can aintain a constant speed of 0.1 m/s?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

25 A wildcat sprinted a distance of 60 meters in 8 s; what was the speed of the wildcat?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

26 If a motorcycle can travels at a constant 40 m/s, how much time will it take the motorcycle to travel 4200 m?

Given Find Formula Manipulate Substitute Answer Diagram

Answer

Another Perspective

Click here for a "minute physics" video on measuring time and distance.

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Constant Speed

Constant Speed

We have just figured out how fast something moves over a given distance and time. How would you know that the object was traveling the same speed from start to finish? — that is, at a Constant Speed?

Constant Speed

Suppose two cars leak oil leaving one drop on the road every second... What can you tell about the motion of the two cars as they go from 0 to 20 meters? Which car is the fastest?.............Why?

Constant Speed

Which car is traveling at a constant speed? What is the characteristic of constant speed? Suppose two cars leak oil leaving one drop on the road every second...

27 A car travels at a constant speed of 10 m/s. This means the car: A increases its speed by 10 meters every second. B decreases its speed by 10 meters every second. C moves with an acceleration of 10 meters every second. D moves 10 meters every second.

Answer

28 Which vehicle is the fastest? A the truck B the car C the bus D the bicycle

Answer

29 Which vehicle is traveling at a constant speed over the 20 m traveled? A the truck B the car C the bus D the bicycle

Answer

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Distance versus Time Graphs

Graphing Motion

So far we have looked at motion using a number line to indicate the distance traveled and the number of dots to indicate the time.

t=0[s] t=1[s] t=2[s] t=10[s]

Another way to look at motion is to make a graph plotting the distance (along the y-axis) against time (along the x-axis).

Graphing Motion

First, we'll make a table from our observations.

t=0[s] t=1[s] t=2[s] t=10[s]

t (s) 1 2 3 4 5 6 7 8 9 10 d (m) 0 2 4 6 8 10 12 14 16 18 20 Now let's graph the data in the table.

Graphing Motion

What would you do differently in this graph?

20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20

Graphing Motion

What would you do differently in this graph?

d(m) t (s) 20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20

Graphing Motion

What would you do differently in this graph?

d(m) t (s) 20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20

Graphing Motion

What would you do differently in this graph?

d(m) t (s) 20 16 12 8 4 0 0 2 4 6 8 10

Graphing Motion

What would you do differently in this graph?

d(m) t (s) 20 16 12 8 4

• 4
• 8
• 12
• 16
• 20

0 1 2 3 4 5 6 7 8 9 10

Creating a Distance vs. Time Graph

· Time and distance are always positive, so the graph will always start with (0,0) at the bottom left as shown. · On graph paper, draw vertical and horizontal axes · Label the horizontal axis (the x-axis) with t, and the unit (s). · Label the vertical axis (the y-axis) with d, and the unit (m).

d(m) t (s)

Creating a Distance vs. Time Graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

Set the "scale" for each axis. · Must start at 0 · Each box should be equal to the same easy amount (1 or 2 or 5 or 10...) so that as much

• f the axis as possible is used to reach the

maximum value · No skipping · Put tick marks where the axis and the side

• f each box cross, and number them. For

the d-axis, we started with 0 where the the axes cross, then put 2 at the next crossing up, 4 at the next and so on.

Creating a Distance vs. Time Graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

Plot the data. What makes this a good graph?

Finding the Speed on a d versus t graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

Draw a straight line through all the points.

Finding the Speed on a d versus t Graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

· The rise is the amount the line goes up between two points. · The run is the amount the line goes across between the same two points. · The slope of the line is the amount the line goes up for every unit it goes across. rise run rise rise run run

Finding the Speed on a d versus t Graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

rise run rise rise run run What is the slope of this line? Hint 1: you can use any two points on the line to find the slope. Hint 2: If (0,0) is on the line, using it as one the points will make it easy to find the slope.

Finding the Speed on a d versus t Graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

rise run rise rise run run Can you see from the units

• f the graph, that the slope
• f a d vs t graph is the

speed?

Finding the Speed on a d versus t Graph

d(m) t (s) 0 1 2 3 4 5 6 7 8 9 10 20 18 16 14 12 10 8 6 4 2

rise run rise rise run run The equation of the line should be familiar to you:

EXERCISE: Our leaky car #1 is still dripping one drop of oil every second.Plot the the motion of the car on the d vs t graph below. What is the car's speed?

t [s] d [m] 3 1 6 2 9 3 12 4 15 5

rise = ____ run = ____ slope =____

EXERCISE: Our leaky car #2 is still dripping one drop of oil every

• second. Plot the the motion of the car on the d vs t graph below.

What is the car's speed?

t [s] d [m] 3 3 6 6 9 9 12 12 15 15

rise = ____ run = ____ slope =____

· Which car was faster: car 1 or car 2? · Which d vs t line had the steeper slope: the one for car 1 or the one for car 2? · What does the steepness of a d vs t line indicate about the speed of the object it represents?

Distance versus Time Graphs and Speed

· Which car was faster: car 1 or car 2? · Which d vs t line had the steeper slope: the one for car 1 or the one for car 2? · What does the steepness of a d vs t line indicate about the speed of the object it represents?

Distance versus Time Graphs and Speed

car 2 car 2 the steeper the d vs t line, the greater the speed

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Bowling Ball Lab

Bowling Ball Lab

When a bowling ball is rolled down an alley or a hallway, · Does it move with an almost constant speed, or a speed that changes a lot? · What is its distance versus time graph? · How fast is the ball moving?

Bowling Ball Lab

When a bowling ball is rolled down an alley or a hallway, · Does it move with an almost constant speed, or a speed that changes a lot? · What is its distance versus time graph? · How fast is the ball moving? In this lab, you will apply what you have learned about · the relationship of d, t, and s · graphing data · determining the slope of a line graph

Bowling Ball Lab

Materials: · Bowling Ball

• 5 Stop Watches
• Meter Stick(s)

· Masking Tape · Ramp (1 meter long board and a stack of 3 textbooks) Setup: · Stick lines of masking tape perpendicular to the intended path at 0, 3, 6, 9,12 and 15 meters. · Place the ramp so that its bottom edge lines up with the tape at 0 meters. · Stick a piece of tape on the ramp board to mark where the board makes contact with the stack of books. · Release the ball from the location on the board marked with tape for consistent results.

Lab Roles: Lab teams will take turns measuring the average speed of a bowling

• ball. For each team:

1. One person (Launcher) will be responsible for releasing the ball. 2. Five people (Timers) with stop watches will line up along the hallway at locations marked with masking tape starting 3 m from the ramp. 3. One person (Catcher) will be located at the far end of the hall to catch the ball. 4. One person (Collector) will collect data for the lab team.

Bowling Ball Lab

Procedure: · Timers take their positions and reset their stopwatches. · The "Launcher" places the ball on the ramp at the tape mark and aims it down the center of the hallway. As he/she release the ball they should signal to the "Timers" to start timing just as the ball reaches the bottom of the ramp. · As the ball passes a "Timer", that person should stop their stopwatch. · When the ball just passes the last tape mark, the "Catcher" will stop it and · The "Collector" will gather the times from each "Timer".

Bowling Ball Lab

Data Collection · record the time measured (t) for the bowling ball to travel the distance from the bottom of the ramp to each tape mark on the floor. · use the speed formula to calculate the speed over each distance. d 3 m 6 m 9 m 12 m 15 m t (measure) s (calculate)

Bowling Ball Lab

Analysis · Plot distance versus time from the data table

Bowling Ball Lab

Analysis · Draw a line of best fit through your data by placing a edge of a clear ruler so that the first point is (0,0) and the rest the points appear to be balanced above and below the edge of the ruler. · Find the slope of this line. This was the "average" speed of the bowling ball over the 15 meters. Looking at speeds you calculated in the table, do you think the ball rolled at a constant speed? Why or why not? How does the average speed you found from the slope compare to these speeds?

Bowling Ball Lab

average speed = _______

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Average Speed

Average Speed

You just calculated the average speed from the plot of distance versus time of a bowling ball. Let's look at average speed in more detail.

Average Speed

Compare the motion of car 1 and car 2... Compare the speeds of car 1 and car 2 during the first

• second. How are they different?

· they have different Instantaneous Speeds (speed at a given moment) But both car 1 and car 2 move 20 meters in 10 seconds... so both have a "speed" of 2 [m/s]. · they have the same Average Speed (savg) (calculated from the distance traveled in a given time)

part 1 part 2 part 3

Average Speed

Where an object has a different speed in each part of a trip, there are a couple of ways to find the average speed, savg. Let's assume a leaky car dripping one drop per second again... Method #1 - just look at the whole trip savg = the total distance traveled ÷ the total time it took to travel that distance.

Average Speed

For the trip below, what are dtotal, ttotal and savg ? dtotal = _____ ttotal = _____ savg = _____ part 1 part 2 part 3

Average Speed

For the trip below, what are dtotal, ttotal and savg ? part 1 part 2 part 3 Each dot represents 1 second of travel time.

part 1 part 2 part 3

Notice that each part

• f the trip has a

different slope, that is - a different speed s. Each s is calculated from the slope from the associated line segment.

20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 d(m)

s1 s2 s3

t (s)

Average Speed

part 1 part 2 part 3

Method #2 - find savg from the d vs t graph · Draw a line from the first point of motion to the last · Find the slope. Notice that savg only depends on the first and last points of the trip.

20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 d(m)

savg = 1 m/s

Average Speed

Average Speed

What is the average speed for the d vs t graph below?

20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 d(m)

Average Speed

What is the average speed for the d vs t graph below?

20 18 16 14 12 10 8 6 4 2 0 0 2 4 6 8 10 12 14 16 18 20 d(m)

savg = 0.7 m/s

Average Speed

Where an object has a different speed in each part of a trip, there are three ways to find the average speed, savg. Method #3 · Find the distance and time for each part of the trip. · Add up the distances of each part of the trip (dtotal) · Add up the times of each part of the trip (ttotal) · Calculate savg

part 1 part 2 part 3

In physics we like to use a shorthand to remind us about what each quantity is about. One way we do this is with subscripts. Subscripts are small notes we write just to the bottom right of the quantity we are working with. For example: if an object makes a three segment trip, we indicate the distances for each segment as d1, d2, and d3 and the corresponding time intervals as t1, t2, and t3 .

Distance and Time Intervals

d1= _____ t1= _____ d2= _____ t2= _____ d3= _____ t3= _____

part 1 part 2 part 3

Average Speed

Method #3 (with physics notation) · Find the distance and time for each part of the trip. · Add up the distances of each part of the trip (dtotal) dtotal = d1+ d2+ d3 · Add up the times of each part of the trip (ttotal) ttotal = t1 + t2 + t3 · Calculate savg

Average Speed - Example 1

Suppose... You ride your bike home from school by way of a friend’s house. · It takes you 7 minutes to travel the 2500 m to his house. · You stay there for 10 minutes · Then you ride the 2700 m to your house in 9 minutes. What was your average speed for this trip? To keep things clear, use a table (graphic organizer) to keep track of the information...

Average Speed - Example 1

Suppose... You ride your bike home from school by way of a friend’s house. 1. It takes you 7 minutes to travel the 2500 m to his house. 2. You stay there for 10 minutes 3. Then you ride the 2700 m to your house in 9 minutes. What was your average speed for this trip? trip segment distance [m] time [min] speed [m/min] 1. 2. 3. dtotal, ttotal savg

• 1. Fill the table with the information given in the story!

Average Speed - Example 1

trip segment distance [m] time[min] speed [m/min] 1. 2500 7 2. 10 3. 2700 9 dtotal, ttotal savg

• 2. Total the distance
• 3. Total the time

Average Speed - Example 1

trip segment distance [m] time[min] speed [m/min] 1. 2500 7 2. 10 3. 2700 9 dtotal, ttotal savg dtotal = 5200 ttotal =26 savg = 200

• 4. Calculate average speed

savg = total distance ÷ total time

Average Speed - Example 2

Suppose... 1. Mike jogs 400 meters at 4 m/s 2. then he walks 2700 meters in 1400 s 3. finally he runs for 100s at 5 m/s What was Mike's average speed on this trip? LEG distance [m] d=st time [s] t=d/s speed [m/s] s=d/t 1. 400 4 2. 2700 1400 3. 100 5 dtotal ttotal savg

÷ =

remember to use the formula

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Position and Displacement

Position

Distance and time didn't require us to define where we started

• r where we ended up. They just measure how far we traveled

and how long it took to travel that far. · When you run or watch a race, do you just want to know the distance traveled and how long it took? · Would you watch if all all you wanted to know is who won at the end? · Would it even matter if the competitors ran at the same time

• r on the same course?

It's the Position of an object (where it is) at a given time that we find interesting!

We are going to start with motion in one-dimension. We will use the symbol x to represent position: left (negative) to right (positive). We also can think about compass directions in terms of positive and negative. Generally we choose: · North as positive and South as negative · East as positive and West as negative · Up as positive and Down as negative

Position in One Dimension

+x

• x

30 All of the following are examples of positive direction except: A to the right B north C west D up

Answer

31 All of the following are examples of negative direction except: A to the right B south C east D down

Answer

Displacement

Now that we have defined position, we can define a change in position. change in position = the final position – the initial position · Displacement is another word for change in position

Every time you want to solve a problem, do you really want to write

• ut things like...?

change in position = the final position – the initial position

Displacement (and some shorthand)

Displacement (and some shorthand)

In physics, we like shorthand, a few symbols to write a lot of words. xf means the final position – or present position xi means the initial position – at the start – when time = 0

Displacement (and some shorthand)

We use the Greek letter ∆ ("delta") for the phrase "change in". so Δx means the change in position ... and that long equation sentence... change in position = the final position – the initial position becomes... Less to remember and less to write!

• x

+y

• y

+x xi (In physics, we label the starting position xi)

Displacement - example

For instance, if you drive 60 miles from Philadelphia, PA to Tom's River, NJ... Displacement describes how far you are from where you started, Regardless of how you got there.

• x

+y

• y

+x xi xf (We label the final position xf) and then 20 miles back toward Philadelphia.

Displacement - example

• x

+y

• y

+x xi xf

Displacement

You have traveled: a distance of 80 miles, but a displacement of 40 miles, since that is how far you are from where you started we can calculate displacement with the following formula:

Displacement

Distance is the number of units (meters, feet, miles...)

• traveled. It is a positive quantity (or magnitude).

If you go out for a 2 mile run as exercise, is the objective to get from one specific place to another, or just to run 2 miles? That's distance!

xf xi

• x

+y

• y

+x xi xf

• x

+y

• y

+x

Displacement

However, displacement can be positive or negative since you can end up to the right or left of where you started.

Displacement is positive. Displacement is negative.

Vectors and Scalars

We distinguish quantities that have: · magnitude (a quantity with units - for example 5 kg or 3 meters) · magnitude and direction (7 meters up or 6 m/s to the left) Scalar - a quantity or number (positive or negative) - physical quantities that have magnitude only are scalars Vector - a quantity that has both magnitude and direction

Vectors and Scalars

Scalar - magnitude only Vector - magnitude and direction

Quantity Scalar or Vector Time Distance Displacement Speed Number of students in this room

Which of the following are vectors? Scalars?

32 How far it is between the starting and ending positions

• f a trip is known as the...

A distance B displacement C magnitude D speed

Answer

33 A car travels 60 m to the right and then 30 m to the left. What distance has the car traveled? A 30 m B 60 m C 90 m D 120 m

+x –x meters Answer

34 A car travels 60 m to the right and then 30 m to the left. What was the car's displacement? (remember left is negative and right is positive) A 30 m left B 30 m right C 90 m right D 90 m left

+x –x meters Answer

35 Starting from the origin (x=0), a car is driven 4 km east and then 7 km west. What is the total distance traveled? (km is short for kilometer or 1000 meters) A 3 km B

• 3 km

C 7 km D 11 km

East West km Answer

36 Starting from the origin (x=0), a car is driven 4 km east and then 7 km west. What is the net displacement from the starting point? A 3 km west B 3 km east C 7 km west D 11 km east

East West km Answer

37 If you run once around a 400 m track, what distance did

you travel? A 0 m B 200 m C 400 m D 800 m

Answer

38 If you run once around a 400 m track, what was your displacement? A 0 m B 200 m C 400 m D 800 m

Answer

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Velocity

Velocity

Velocity is to Speed as Displacement is to Distance

Velocity

Speed is defined as the ratio of distance and time Similarly, velocity is defined as the ratio of displacement and time Note: we use t as the time interval or amount of time taken, not as the clock time! velocity = time elapsed displacement speed = distance traveled time elapsed

Velocity

Speeds are always positive, since speed is the ratio of distance and time; both of which are always positive. But velocity can be positive or negative, since velocity is the ratio

• f displacement and time; and displacement can be negative or

positive. Remember... Left (negative) Right (positive)

39 Which of the following is a vector quantity? A time B velocity C distance D speed

Answer

40 Velocity is defined as change in ______ over a period

• f ______.

A distance, time B distance, space C displacement, time D displacement, space

Answer

41 You travel 60 m to the right in 20 s; what is your velocity? A 3 m/s B 3 m/s right C 1200 m/s right D 3 m/s left

Answer

42 You travel 60 m East in 20 s;what is your velocity? A 3 m/s B 3 m/s left C 1200 m/s East D 3 m/s East

Answer

43 What is your average speed when you run once around a 400 mtrack in 80 s? A 0 B 5 m/s C 0.2 m/s D 32,000 m/s

Answer

44 What is the magnitude (without direction of your velocity when you run once around a 400 m track in 80 s? A 0 B 5 m/s C 10 m/s D 2.5 m/s

Answer

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Position-Time Graphs

Position vs Time Graphs

Position versus Time graphs are just like Distance vs Time graphs except: · position can go backwards · position can be positive

• r negative.

Position vs Time Graphs

On the graph: · change the label on the vertical axis (y- axis) to x (m) - for position · time still starts at 0 · you might need to let the vertical axis go negative x (m) t (s)

x (m) t (s) – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – –

Position vs Time Graphs

With distance vs time, we assume motion starts at d = 0 when t = 0. With position vs time, motion starts at position xi which can be anywhere. For example, if you

• bserve all the cars

parked on your street at noon, they would all have different positions at that moment. xi

x (m) t (s) – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – xi moving forward position increases not moving position stays the same moving backward position decreases and may become negative

Position vs Time Graphs

With distance vs time, the distance traveled will never become less. (you can't undo how much you walk!) With position vs time, an

• bject can stay still,

move forward (+) or backward (-). For example, if you

• bserved all the cars

parked on your street at noon, some may move to the right, some to the left, and some may not move at all.

Position vs Time Graphs

With distance vs time, the slope of the line was the speed: With position vs time, the slope is the velocity: x (m) t (s) – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – xi

1 1 2 3 4 5 6 7 8 9 10 11 2 3 4 5 6 7 8 9 10111213

∆x = 10-2 = 8 m t = 12 s

Position vs Time Graphs

With distance vs time, the slope of the line was the speed: With position vs time, the slope is the velocity: x (m) t (s) – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – xi

1 2 3 4 5 6 7 8 9 10 11

• 1
• 2
• 3
• 4
• 5
• 6

1 2 3 4 5 6 7 8 9 10111213

∆x = -2-(-6) = -8 m t = 8 s

x (m) t (s) – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – – xi moving forward slope > 0 v >0 not moving slope = 0 v =0 moving backward slope < 0 v < 0

Position vs Time Graphs

Speed and Velocity Summary

Speed Displacement Velocity