AP Physics1: Kinematics 4: Dot-Timer and Motion Graphs

Summary

This lesson introduces the concept of motion and how to record it using a low-tech tool called a ticker timer or dot timer. The video explains how the timer marks dots on paper tape as a low-friction cart moves down an incline, allowing us to measure the cart's position over time. Viewers learn to calculate average speed, instantaneous speed, and velocity using the data from the dots. The lesson also covers how to plot a position vs. time graph, find displacement, and understand how the area under a velocity-time graph represents displacement.

Takeaways

• 🛠️ The lesson introduces tools to record motion, with a focus on using a low-tech device called a ticker timer (or dot timer).
• ⚙️ A ticker timer consists of a motor that causes a metal blade to vibrate, producing dots on a piece of paper tape dragged by a moving object.
• 📉 The dots on the paper tape represent the object's motion, spreading out more as the object speeds up, providing a visual representation of acceleration.
• 🧮 The distance between dots on the tape helps quantify motion, and measurements of the distance from the origin provide the position of the moving object at different times.
• ⏲️ The ticker timer produces 60 dots per second, allowing precise time intervals between measurements (e.g., four intervals = 4/60 of a second).
• 🚗 The lesson focuses on using the ticker timer to analyze the motion of a low-friction cart moving down an incline, correlating time intervals and distance traveled.
• 📏 Average speed is calculated by dividing the distance traveled by the elapsed time, and instantaneous speed can be determined by focusing on smaller intervals.
• 📊 A position vs. time graph reveals the curve of the motion, showing increasing distance traveled as time progresses and the cart accelerates.
• 📈 The instantaneous velocity at a specific time can be determined by finding the slope of the tangent line on the position vs. time graph.
• 📐 Displacement is represented as the area under the curve on a velocity vs. time graph, and smaller time intervals provide more accurate approximations.

Q & A

What is the purpose of using a ticker timer in the experiment?

The ticker timer is used to record motion by creating dots on a piece of paper tape. These dots provide data that can help describe the motion of an object, such as a cart rolling down an incline.

Why is the term 'DOT timer' used to describe the ticker timer?

The term 'DOT timer' is used because the device produces dots on the paper tape as it taps up and down. These dots represent specific intervals of time, helping to record the motion of the object.

What is the significance of the dots spreading out as the cart rolls down the incline?

The dots spread out farther apart as the cart speeds up. This shows that the cart's velocity increases over time as it moves down the incline, which can be measured to analyze the motion.

Why are the first few dots close together on the tape?

The first few dots are close together because the cart starts moving slowly at the beginning of its motion. Additionally, the paper tape may have been curled or straightened out, contributing to the closer dots.

What information is provided by marking every fourth dot on the tape?

By marking every fourth dot, the experimenter averages out the possible unevenness of the dot timer's intervals. This provides more consistent data for measuring the cart's position at different time intervals.

How is average speed calculated from the ticker timer data?

Average speed is calculated by dividing the distance traveled by the time taken. For example, the distance between two points on the tape divided by the time difference between the two points gives the average speed in cm/s.

What is the difference between average speed and instantaneous speed in this experiment?

Average speed refers to the total distance traveled divided by the total time taken, while instantaneous speed is the speed at a specific moment in time, calculated by taking measurements over a very short time interval.

How is the instantaneous velocity determined from the position versus time graph?

Instantaneous velocity is determined by finding the slope of the tangent line to the position versus time graph at a specific point. The slope represents the rate of change of position, or velocity, at that moment.

What does the shape of the position versus time graph indicate about the cart's motion?

The position versus time graph shows a curve that curves upwards, indicating that the cart is accelerating as it moves down the incline. The increasing distance between dots on the tape corresponds to this acceleration.

How is displacement related to the area under the velocity versus time graph?

Displacement is equal to the area under the velocity versus time graph. The area represents the total change in position (displacement) over the time interval. This can be calculated by summing the areas of small rectangles under the curve.

Outlines

00:00

🔧 Introduction to Tools for Recording Motion

This paragraph introduces the concept of motion and the tools used to record it. While modern technology offers advanced tools like GPS and video recordings, the lesson focuses on a simpler, low-tech tool called a ticker timer (or DOT timer). The paragraph describes the ticker timer's functionality: it vibrates a metal blade with a pointy screw, which creates dots on a paper tape passing under carbon paper. The paragraph sets the stage for using this tool to track the motion of a low-friction cart.

05:01

📏 Measuring Motion with the DOT Timer

The second paragraph explains how to use the DOT timer to measure motion. The cart’s motion is recorded as dots on the paper tape, with the dots spreading farther apart as the cart accelerates down an incline. It discusses marking every fourth dot to account for potential timer inaccuracies and establishing an origin point for measurement. The position of each dot is then recorded, and a process is outlined to measure the distances from the origin to each dot, leading to the collection of data for describing the cart's motion quantitatively.

10:02

⏱️ Calculating Average Speed and Velocity

Here, the script delves into calculating the average speed and velocity of the cart. By using the measured positions of the cart at specific time intervals, the paragraph describes how to calculate average speed, which is defined as the distance traveled divided by the time. It highlights the importance of understanding that, since the cart moves in a straight line, the average speed and average velocity are the same. An example calculation is provided, resulting in an average speed of 48.5 cm/s.

15:02

⚡ Instantaneous Speed and Graphing Motion

This section shifts the focus to instantaneous speed, calculated over very short time intervals. It suggests measuring the distance between two dots, then dividing by the corresponding time interval. The example gives a speed of 55.5 cm/s. The paragraph also introduces graphing the motion, predicting the shape of the position vs. time graph. The curved line represents increasing speed as the cart accelerates, and a smooth curve is drawn to fit the data points for better accuracy.

20:03

📉 Slope as a Measure of Instantaneous Velocity

In this paragraph, the concept of the slope is used to describe instantaneous velocity. The position vs. time graph is used to determine instantaneous velocity at any given moment by finding the slope of a tangent line to the graph. The slope, or rise over run, represents the velocity at that point. This method is also linked to finding acceleration using the velocity vs. time graph, with acceleration represented by the slope of that graph.

📐 Displacement as Area Under the Curve

The final paragraph explores how displacement is represented by the area under the velocity vs. time graph. By dividing the graph into small rectangles, the area of each can be added together to estimate the total displacement over time. The more finely the time is segmented, the more accurate the calculation becomes. This approximation helps visualize the relationship between velocity and displacement, laying the groundwork for solving graph-based problems in future lessons.

Keywords

💡Ticker Timer

A ticker timer is a device used to record motion by making marks on a tape at regular intervals. In the video, the ticker timer is connected to a power outlet and is used to measure the motion of a cart by creating dots on a carbon paper as the cart moves. It helps visualize and analyze the motion over time.

💡Dots

The 'dots' refer to the marks made by the ticker timer on a piece of tape. These dots represent the position of the moving object (the cart) at equal time intervals, allowing for the analysis of motion. As the cart moves faster, the dots spread farther apart, illustrating the increasing speed of the cart.

💡Low Friction Cart

The low friction cart is an object that rolls down an inclined plane with minimal resistance, used in this video to demonstrate motion. The tape attached to it records the cart's movement, showing how it accelerates as it moves down the incline, providing data to measure speed and displacement.

💡Position

Position refers to the location of the cart at a given time, measured from the origin (starting point). In the video, the position is determined by measuring the distance between the dots on the tape, helping to calculate the cart's displacement and velocity over time.

💡Time Intervals

Time intervals represent the equal time gaps between each dot on the tape created by the ticker timer. Since the ticker timer makes 60 dots per second, each interval is 1/60th of a second. These intervals allow for the calculation of both the average speed and instantaneous velocity of the cart.

💡Average Speed

Average speed is defined as the total distance traveled divided by the total time taken. In the video, it is calculated by dividing the change in the cart’s position between two dots by the time difference. Since the motion is straight, the average speed equals the average velocity.

💡Instantaneous Speed

Instantaneous speed refers to the speed of the cart at a specific moment in time. It is calculated by measuring the distance between two nearby dots and dividing it by the time interval. This gives a more accurate idea of how fast the cart is moving at that exact moment in its motion.

💡Position vs. Time Graph

A position vs. time graph is a graphical representation of an object's position over time. In the video, the graph curves upward, indicating that the cart is accelerating. The graph helps visualize how the cart’s position changes as it moves faster down the incline.

💡Tangent Line

A tangent line is a straight line that touches a curve at a single point without crossing it. In the context of the video, the tangent line on a position vs. time graph represents the instantaneous velocity at a given moment, as its slope corresponds to the velocity at that time.

💡Displacement

Displacement refers to the change in position of the cart from its initial to its final point. In the video, displacement is calculated using the area under the velocity vs. time graph, illustrating how far the cart has moved over a certain period.

Highlights

Introduction to describing motion and the tools used for recording motion, focusing on low-tech solutions like the ticker timer.

Explanation of how the ticker timer works, including its mechanical parts such as the motor, vibrating blade, pointy screw, carbon paper, and paper tape.

Demonstration of how the ticker timer creates dots on paper tape as a way to record the motion of an object, such as a low-friction cart.

Prediction that as the cart picks up speed while rolling down an incline, the dots on the tape will spread out farther apart, illustrating acceleration.

Emphasis on using a marked x-axis to measure the distances between dots to describe the motion quantitatively.

Details on how time intervals between dots are measured, with the ticker timer creating 60 dots per second and measurements taken every four dots.

Clarification that average speed is equal to average velocity in this scenario, as the motion is in a straight line.

Explanation of how to calculate average speed by dividing the total distance by time, using the data from the ticker timer.

Description of how instantaneous speed can be calculated by measuring the distance between two dots and dividing by the time interval.

Introduction of graphing motion, plotting position versus time and predicting a curve due to the increasing distance between dots as the cart speeds up.

Instruction on drawing a smooth curve for a best-fit line on the graph, even when experimental data may not be perfectly aligned.

Discussion of how instantaneous velocity can be found by calculating the slope of the position versus time graph at a specific point.

Introduction to the concept of displacement being equal to the average velocity multiplied by time, especially in relation to velocity versus time graphs.

Explanation of how the area under the velocity versus time graph represents displacement, with smaller time intervals improving the accuracy of the approximation.

Summary of how to interpret and analyze graphs, finding the instantaneous velocity and displacement using slopes and areas under the curve.

Transcripts

now we have learned the terms we use to

describe motion it can be nice to have

some tools to help us to record motion

so we can have data to use to describe

motion more carefully you can probably

think of many different ways to record

motion especially because we have easy

access to so much technology these days

we can use a video recording or GPS for

motion over long distances but for this

particular lesson for starters we'll be

using something that is pretty low Tech

this thing

here is called a ticker timer or a DOT

timer and you plug for this particular

one you can plug it into a wall

outlet and then when you plug it in this

motor here can be turned down and then

the motor is going to make this metal

blade over here vibrate up and down at

the end of the blade there's a pointy

screw attached to it which means that

that screw would a tap up and down this

one here is a circular piece of a carbon

paper underneath carbon paper there's a

paper tape piece of paper tape going

down let me plug it in okay let me turn

the switch

on see it makes that sound that's why

some people call call it a tick

timer when it's on this pointy screw

would hit

the carum paper

repeatedly so what do you think happens

on that piece of

paper yeah you get dots of course in

this case all the dots are at the same

place since it makes dots that's why we

call it a DOT

timer now I want to use this that timer

to record the motion of this low

friction cart rolling down this

incline how should I do

it yep I can attach tape to the

card turn on the timer and release the

card as the card drags the tape with it

the dots on the tape would spread

out now think about the motion the card

went through what do you expect that to

look like on the

tape that's right the dots on the tape

should spread out and they should spread

out farther and farther part as the card

picks up speed down the incline now

let's look more carefully at these dots

and make some

measurements now you can get a closer

look at the dots on the tape although

they are kind of too light for you to

see clearly so I'm going to darken them

for

you now let's use these dots to describe

the motion of the card

quantitatively since it is a

one-dimensional motion we can use a

x-axis

conveniently notice how I Mark the dots

every four

spaces 1 2 3 4 1 2 3 4 the reason why I

did not Mark every dot is that the dot

timer I used may have an even time

intervals between dots by taking data

every four intervals I'm hoping to

average out the

unevenness I begin the first dot over

here because these dots come before for

it are really close together remember I

turned on the timer before releasing the

card so the first few dots overlap the

paper can also be curled at beginning so

these ones may be caused by the paper

tape getting straightened out by the

card and then of course when the card

first started it's very slow so the dots

naturally will be close

together so in any case I'm starting my

measurements from this St right here

which means that I can conveniently make

it my

origin xals to zero right here and then

if I want the position for this one and

these other ones I'll just have to find

their or x coordinate which means I need

to find the distance to the origin so if

I make this measurement between the

first dot origin and this one here it

will be about 1.3

cm and then for the second one I will

also need to measure all the way to the

origin because the x coordinate is the

distance to the origin so for all of

these I have to go measure the distance

to the origin okay so I'm going to make

those measurements so now I have x = 0

and then this x = 1.3 CM 3.6 6.8 11 and

6 16.1

cm of course I also need the time so I

can say at the beginning right here t

equal to

Zer and then for this it's a four spaces

for that dot timer we used it makes 60

dots every

second 60 dots every second means every

time interval between the dots is 160 of

a second so 1 2 3 4 four intervals means

when the start is made the time will be

460s of a second and that one

860s of a

second and so on and so

forth so I labeled the time I can ask

you say what is the position of the cart

at T T = to

1660s of a

second then you would measure the

distance all the way to the origin and

that will give you 11

cm I can ask you questions about average

speed now in this case the average speed

and average velocity will be the same

thing because the card went straight no

zigzagging no curving around

so the average speed will be kind of

like average velocity which means

distance traveled and displacement will

be the same let's say if I want to ask

you to find the average speed which is

the distance travel divided by the

time

from T = to 460s of a second to 1660

of a

second then let's see we'll need the

distance traveled between this time and

that and that will mean it's 11us

1.3 divided by the time final time minus

the initial time that will be

1660 minus

460 so that is 9.7 /

1260s which means it's 9.7 * 60 / 12 9.7

* 5 so this gives you

48.5 and what do you think the unit is

since we used centimeters for the

distance seconds for the time this is

going to

be cm per second so that's the average

speed average velocity will be the same

amount now of course our distance the

standard unit is meters but for this

scale centimeters is convenient so it's

perfectly fine to use

centimeters what if I ask you about

instantaneous speed so

let's see I can ask you what is the

instantaneous speed at T equals to say

1260s of a

second now remember to find the

instantaneous speed you would have to do

the average as well but you have to take

your average over a very short amount of

time so let's come here if you want the

average speed around here the very short

amount of time you have a choice you can

measure the distance

between two dots one time

interval but uh since the speed is

changing it may be a good idea to do a

little bit at before and a little bit

after and if you want to consider about

evening out

the time because of the unevenness of

the timer you can go two dots before two

dos after three before three after or

four before four after any of these will

work but basically if you want the

instantaneous speed you want to take

average over a short amount of time okay

so let's say if I choose to measure the

distance between these two

dots I'll have to measure the distance

and then divide it by the time interval

the distance between these two dots will

be three 3.7

CM so this will

be

3.7

CM divided by the time interval from

here to here again that will be four

spaces so it'll be four 60 of a

second and if you do this calculation

you're going to get

555 and again that would be cm per

second

so that will be the instantaneous

speed

at 1260s of a second right over

here as in many fields we can show

people the data we have in numbers we

can also show our data in graphs for

different effect so let's plot a

position versus time graph which means a

X versus T

graph before actually plotting it

see if you can figure out what shape

graph to expect by looking at how dots

spread

out so you probably have figured out

that once you plot the dots on the graph

in this case I have CM for the x axis

and times 1 16 60s of second for the

horizontal axis the

time I get a curve that's curving up

like this not a straight

line okay because as you can see the

death distance traveled by the card gets

bigger and bigger for the same time

interval because the C speeds up as it

goes down the

incline and that means that in the same

time interval the distance TR

traveled between the interval gets

bigger and the

bigger and

bigger in physics we like to we don't

connect the dots but we like to make a

best fit curve so I'm just going to very

carefully draw a smooth curve that

fits as many dots as

possible okay here's my best fit curve

and in this particular case I happen to

get a curve that's uh very smooth and

fit the dots very

well most of the time when you carry out

an experiment because you have you have

errors therefore the dots may not fit

the smooth curve very well you may have

some dots that's higher some dots are

lower but that's

okay what that's what we expect from

experimental

data but in any case physicists like to

make smooth curve we don't really like

to have dots connected with straight

lines with this smooth curve we'll be

able to find the position of the

object at different moment for example

we can find the position at

16 60s of a second and then read it off

the graph and figure out the X position

we can also find the position of the

cart at

1460th of a second we can just read off

the graph along that smooth curve we

have if we want the average velocity

from a certain moment to another

moment of course we can do that too

because we can find the positions find

the change in position or the distance

traveled in this case they're the same

numbers divide by the

time what if we want the velocity at a c

certain moment for example we want to

find the velocity at T = to 1260 of a

second in that case we'll have to do the

same thing for

instantaneous velocity or instantaneous

speed what we want is the distance

traveled divided by time or displacement

traveled divided by time for the

instantaneous is velocity which means

we'll still need to look at the distance

traveled divided by the

time but we'll have to do that over a

very short amount of time so let's say a

little bit before and a little bit

after so let's say we are looking at

this little

segment a little bit before and a little

bit after a little a little bit before

it's X position is here a little bit

after it's X position is right

there which

means your distance traveled or the

displacement would be Delta X that is

your Delta

X and what is your delta T this is your

initial time that that is your final

time so this part is your delta

T which means to find the instantaneous

velocity at t equal to

1260s of a

second we will have

to find the distance travel divided by

the time for a very short amount amount

of time so in this case it's the Delta X

over delta T that means it's a rise over

run so if I make a tangent line a line

that is the tangent to My

Graph this distance

traveled divide by the

time would be the rise over run which is

the

slope of this line that's tangent to the

X versus T

graph instantaneous velocity would be

the slope of this graph so we can write

it over

here the instantaneous velocity is the

slope of the position versus time

graph similarly if we're talking about

instantaneous acceleration since it is

an average Delta V over delta T over a

very short amount of time that means

that this would be also the slope but

this will be the graph that is velocity

versus time graph because your Delta V

is the rise the delta T would be the run

so the rise over

run Delta V over delta T that's the

slope of the graph if you're dividing

that is slope because it's rise over run

rise over

run since the rise is Delta V the

vertical axis is V since the run is

Delta T the horizontal axis is the

T now let's go over the last thing we

need to add on this terms table that is

the

displacement now according to this part

you can see that displacement is the

average velocity times time it is the

average velocity times time and it

equals to something let's see let's look

at this velocity versus time graph

and let's find the displacement

corresponds to this

motion and this should be the average

velocity times time but the velocity

keeps changing so it's kind of hard to

tell what the average velocity is but we

can do this if we chop the time into

little

segments say shorter amount of time

delta T then the velocity doesn't change

so much so we can say the average

velocity is about this much so this will

be the average velocity and then I

multiply this by the time that's gives

that that is the height times the base

which gives you the area of this

rectangle so you can do all these

different rectangles add all the areas

together and you will get the total

displacement for whatever time range

you're looking at now of course this is

not quite so accurate to make it more

accurate what we can do is to chop the

rectangles to thinner pieces then you

get a better

approximation and then if you make it

even thinner it's getting even closer

now if you make them extremely thin it's

going to be pretty

accurate and that means when you make

them extremely

thin the rectangles added together will

give you the area under this curve the

under the area under the Curve will be

the

displacement so that's our last part the

displacement would equal to the area of

a graph it's the area of the Velocity

versus time graph see the area of a

rectangle is the height times the

base this is the height that's the base

when you are multip multiplying that's

referred to the area and that's the

height is the velocity so the vertical

axis is the velocity the base is delta T

so the horizontal axis must be the time

so it's the height times the base that's

the area that's the

displacement in our next video we will

practice solving some graph problems