Intro to Motion
My partner and I were asked to brainstorm some ways of describing motion. Off the tops of our heads, we thought of:
-Is there a pattern to the object's movement?
-What path does it take?
-What forces are acting on the object's movement?
-What is its rate of movement? (how distance does it cover in a specific amount of time?)
-Does it change directions?
These ways of describing motion are based on observation alone. By analyzing the data we gather about an object's change in distance and change in time we can understand the object's change in position.
A good way to represent data gathered form observing an object is by a graph.
Some quick definitions: Qualitative data: data that shows general trends or patterns, not individual amounts. Quantitative data: data that shows accurate amounts and measurements.
We have been discussing graphs in class over the past week. From our discussions, I have gathered that:
-a motion map is an effective method of physically representing the motion of an object. The distance is established, and the object is drawn at intervals to represent where it is at certain times. -Advantage: actually shows you the object's path. -Disadvantage: Not as exact or quantitative as it could be. A motion map is more qualitative, meaning that it gives you the gist of things.
-an x-t graph is a more quantitative way of representing data. When representing movement, units of measurement are given for distance (typically x-axis) and time (typically y-axis). The slope of the data, or the difference between the points, is the speed. -Advantage: Data can be quantitatively displayed. -Disadvantage: If the data being displayed isn't perfectly linear, the slope of the entire data represents average speed instead of the actual changing speed. Also, the dots of data plotted on an x-t graph do not actually show you the objects path--for example, a graph with a diagonal line going upwards to the right does not mean that the object itself moved upwards and to the right, but that the object increased in speed as time when by. X-t graphs show changes in position and distance, not the actual path of the object, and this can be confusing to someone with little experience reading graphs or interpreting data. Examples (graphs created by me, with the aid of this website and Excel):
Graph A: Constant Speed
Preview of your graph
The object portrayed in this graph starts at zero seconds, zero meters. Once in motion, it goes five meters per one second--meaning that it's slope is 5 meters over 1 second. This also means that the speed of the object, which is the same as the slope, is 5 meters per second.
The difference between the plotted points is consistently 5 over 1, meaning that the slope and therefore the speed of the object is constant. Its shape is due to the fact that the object moves the same number of meters each second as time goes by.
In this graph, the object in motion increases by changing amounts. From 0 to 1 seconds, the object moves 0.5 meter. From 1 to 2 seconds, the object moves 1 meters, and from 2 to 3 seconds, the object moves 1.5 meters. The distance between the rest of the points increases by 0.5 meters for each second. From this we know that the slope is not staying the same over time, but increasing, meaning that the speed of the object is increasing over time as well. The shape of the graph is due to the fact that the object is moving more meters per second as time goes by.
Graph C: Decreasing Speed
Preview of your graph
Preview of your graph
In this graph, the difference between 1 and 2 seconds is 4 meters. The difference between 2 and 3 seconds is 3.5 meters, and the difference between 3 and 4 seconds is 3 meters. The difference between the points decreases by 0.5 meters each seconds. From this we can see that the slope is decreasing as time passes, meaning that the speed is decreasing as well. The shape of the graph is due to the fact that as time goes by, the object moves fewer meters each second.
Another method of visually representing motion is through a v-t, or velocity-time, graph.
Velocity is the rate of change of position of an object, or its change in speed over time.
This graph reflects a v-t version of the x-t graph above of constant speed. The x axis is time in seconds. The y axis is meters per second, which, on this graph, remains consistently at 5 meters per second.
On an x-t graph, a horizontal line of data would suggest that no change in distance occurred, and that therefore no motion occurred and that there was no speed. On a v-t graph, this means that the object's velocity did not change, but stayed constant, which means that the speed is increasing at a constant rate. Through this we see that similar shapes does not always mean similar information on different types of graphs.
This v-t graph represents the change in velocity for the data recorded in the x-t graph of increasing speed shown above. The x-axis is seconds, and the y axis is meters per second. Instead of curving as the x-t graph does, this graph is in a diagonal line that does not curve because the velocity is increasing at a constant rate.
On an x-t graph, data in this shape would suggest that the object was moving at a constant speed, but on a v-t graph, the data shows that the velocity of the object was increasing at a constant rate. Despite the differences in their shapes, both graphs accurately portray the same object's motion.
Note to Mr. Todd: Some of the graphs are from a graph-making website, but it got all gross and glitchy, so some of them are represented by Excel documents.
Relative Motion
We had a class assignment to create a video portraying relative motion and think of some "deep" questions to ask the class upon presentation.
This is our video:
Brought to you by Andrea Tate, Elly Bryant, Alley Bowers and myself.
The questions are as follows:
1. What motion do you see?
2. How does the motion differ from the person in the background to the two spinning girls?
3. Describe the motion of the spinning girls in relation to each other and in relation to the creeper in the background.
After viewing and discussing the situation with the class, it was decided that:
1. Two of the girls are spinning in a single circle, while one is walking in a straight line, getting closer to them for the first portion of the video, and then passing them and getting continually farther away.
2. Their motion differs in that the spinning girls travel in the same path for the duration of the video, the walking girl's path takes her forward onto new ground.
3. Relative to each other, the spinning girls are not moving because the distance between them is not changing as they spin. The girl in the background, however, gets closer and then farther away to the spinners, changing the distance between them and therefore moving relative to them.
This Thing You Call "Relative Motion"
So we did the project and whatnot, but what does it mean? And why does it matter?
Well, relative motion is defined as the motion of an object relative to another object.
For motion to occur, there must be something to compare the motion to--otherwise, how could you possibly know that it was moving? Think about how we describe a moving object--it got closer to me. It went away from the bus. For motion to occur, there has to be a change in distance, yes? And to measure this distance, we need a zero. This is the object/point in space that the object in motion is moving relative to. On a graph, it is the zero point (aptly named, this zero).
For example:
There are two cars parked next to each other in a parking lot, one red and one white. The red car starts up and drives forwards, away from the white car. Relative to the white car, the red car is moving, because the distance between them is changing. In this scenario, the white car is the zero and the red car is the object in motion.
But wait. If the definition of motion is a change in distance, wouldn't the white car be moving relative to the red car, too, even if it isn't the one driving away?
Yes. Exactly.
If you were to use the red car as the zero and the white car as the object in motion, the white car would be moving relative to the red car. In every situation, if object a is moving relative to object b, then the opposite is true as well. It just depends on where you place your zero.
So, keeping these ideas in mind, let's set up another scenario.
Now the red car and the white car are both driving down the road, the white one behind the red one, both going 35 miles per hour.
Because of their identical speed, the distance between them is not changing--but they're both being driven. Are they moving, relative to each other?
No.
The same rule still applies. If there is no change in distance between them, then, relative to each other, neither car was moving.
HOWEVER. This does not mean that neither car was moving--relative to other things.
Think of all the things that the cars' distances are increasing or decreasing from--the point in the parking lot at which they began driving, a sign that they pass by on the road, a hitchhiker left in the dust. So many possibilities.
Bottom line: If the distance is changing between objects, then they are both moving relative to each other.
And...why does it matter? Because all motion is relative. Without an object to compare it to (a zero), it is impossible describe (much less perceive) the motion of another object. Graphs cannot be made. Measurements cannot be taken. And the nice thing is, if you've ever described the motion of anything, you've been using the idea of relative motion your whole life without even realizing it.
How does it affect science?
To quote from this website:
"Motion means continuous change of position of an object with respect to an observer. To another observer in a different frame of reference the object may not be moving at all, or it may be moving in an entirely different manner. The motions of the planets were found in ancient times to appear quite complicated...By transferring to the frame of reference of an imaginary observer on the Sun, Johannes Kepler showed that the relative motion of the planets could be simply described in terms of elliptical orbits. The validity of one description is no greater than the other, but the latter description is far more convenient."
Basically, it helps us to better understand the motion of objects by being able to use different reference points to perceive it.
Parallax
I am looking at a flower vase at the end of the next room, perhaps fifteen or twenty feet away.
It appears to be right in the middle of my line of vision.
Viewing with left eye: The vase seems more left-ish.
Viewing with right eye: The vase seems much more to the right.
Switching back and forth between using one eye or the other, but never both at one time, the vase appears to be jumping from left to right and right to left, depending on the eye. But the vase isn't actually moving, is it?
This phenomena can be described as parallax, "the apparent displacement of an observed object due to a change in the position of the observer." (link)
Think about it. You aren't a cyclops. Your eyes are positioned a small space away from each other. When observing with both eyes, we see an object in a single place, but when we switch between using one eye or the other, the apparent difference in the position of the object shows that each eye percieves it from a different angle.
On a larger scale, you could stand to the right of a chair. The chair, obviously, appears to be at your left.
Move behind the chair and it appears to be directly in front of you.
Move to the left of the chair, and now it is on your right.
Just like before, the observed object has not changed its position, but the observer has.
However, distance makes a difference as to how you perceive an object. If I do the same thing that I did to the vase with the house outside of the window, all the way across the street, it doesn't seem to "move" at all. Why? My eyes are still perceiving the house from different points, right?
Fact is, the farther away an object is, the less its position will seem to change. Relative to the distance between my eyes and the house, the distance between my eyes is very small. To notice any change in postion, I would have to view the house from two points that are farther apart, like from one end of the block to the other. Likewise, the greater the distance is between the points of observation, the more the position of the observed object will appear to have changed.
So, when observing an object close to you that is aligned with a distant object and observing it from each individual eye, the close object will appear to move a lot in comparison to the distant object. They will no longer be aligned. This fact is particularly useful to astronomers.
How is parallax used?
Astronomers can use parallax to determine the distance to stars. It might seem like a far stretch from looking at a vase in your living room, but the principal is the same.
using motion parallax to measure distance to a nearby star
using motion parallax to measure distance to a nearby star
(Info and image pertaining to astronomy obtained from this website. Anything in quotations is text taken directly from the site.)
In this case, a person's eyes are not far enough apart to experience parallax when looking at a star. A much greater distance is needed--like the diameter of the earth's path around the sun, as shown at left.
"The parallax angle, P, is measured by comparing the nearby star's position to the stable position of distant background stars."
Basically, the astronomers use the known characteristics of triangles to determine the angles and therefore distances of the triangles created by the points of observation, the star being observed, and the distant stars being used as background reference points.
Using the operations of sine, cosign or tangent, if one of the other angles of the triangle is known (like the distance from the earth to the sun) and the angles have been found, the distance from the earth to the star is able to be calculated.
The Intersection
THE SITUATION:
One way to test one's knowledge of velocity is to apply it to a situation.
Let's say we have an intersection. Car A is stopped at a stop sign at the intersection's edge, waiting for its chance to go. The cross-traffic is traveling on the street perpendicular to Car A at a constant speed of 3 meters per second. All cars are 4.6 meters long.
The distance across the intersection, from curb to curb, is 14.23 meters.
The distance from the stop sign to the opposite curb is 19.4 meters.
The closest car in the cross traffic, Car B, is right with its front at the edge of the intersection and is going 3 meters per second. Car A is positioned with its front right at the stop sign. It is stopped, but has an acceleration of 2.237 meters per second per second.
Starting the cars at these positions, who will get across the intersection first? How long does it take each vehicle?
THE FORMULA:
These formulas can be used to figure the time an object takes to cover a certain area, change in position and velocity being known.
change in distance=(1/2 acceleration x time squared) + (initial velocity x time)
or change in X=1/2at^2 + Vit
also:
Change in distance=1/2 velocity x time
or change in X=1/2V x t
THE STEPS:
Taking the information given, these immediate conclusions can be made:
- Car A will have 24 meters to travel (length of car+distance from stop sign to intersection, 4.6+19.4).
-Car B will have 18.83 meters to travel (length of car+distance form one side of the intersection to the other, 4.6+14.23)
The distance of the car must be taken into account because the car is not fully out of the intersection until its end has passed the curb on the opposite side.
Car A:
change in X= 24
acceleration= 2.237
initial velocity= 0
Time is the variable that needs to be found.
24=1/2 2.237t^2 + 0t
24=1.1185t^2 + 0
21.457=t^2
t=4.632
The time it takes Car A to reach the other side of the intersection from the stop sign is 4.632 seconds.
Car B:
change in X= 18.83
velocity= 3
Here, again time needs to be found.
18.83=1/2 3 x t
18.83 = 1.5 x t
12.553 = t
The time it takes Car B to cross from one side of the intersection to the other is 12.553 seconds.
THE ANSWER:
Car A will reach the other side of the intersection before Car B, even though Car A has a larger distance to cover. This is due to its acceleration.
Intro to Motion
My partner and I were asked to brainstorm some ways of describing motion. Off the tops of our heads, we thought of:
-Is there a pattern to the object's movement?
-What path does it take?
-What forces are acting on the object's movement?
-What is its rate of movement? (how distance does it cover in a specific amount of time?)
-Does it change directions?
These ways of describing motion are based on observation alone. By analyzing the data we gather about an object's change in distance and change in time we can understand the object's change in position.
A good way to represent data gathered form observing an object is by a graph.
Some quick definitions:
Qualitative data: data that shows general trends or patterns, not individual amounts.
Quantitative data: data that shows accurate amounts and measurements.
We have been discussing graphs in class over the past week. From our discussions, I have gathered that:
-a motion map is an effective method of physically representing the motion of an object. The distance is established, and the object is drawn at intervals to represent where it is at certain times.
-Advantage: actually shows you the object's path.
-Disadvantage: Not as exact or quantitative as it could be. A motion map is more qualitative, meaning that it gives you the gist of things.
-an x-t graph is a more quantitative way of representing data. When representing movement, units of measurement are given for distance (typically x-axis) and time (typically y-axis). The slope of the data, or the difference between the points, is the speed.
-Advantage: Data can be quantitatively displayed.
-Disadvantage: If the data being displayed isn't perfectly linear, the slope of the entire data represents average speed instead of the actual changing speed. Also, the dots of data plotted on an x-t graph do not actually show you the objects path--for example, a graph with a diagonal line going upwards to the right does not mean that the object itself moved upwards and to the right, but that the object increased in speed as time when by. X-t graphs show changes in position and distance, not the actual path of the object, and this can be confusing to someone with little experience reading graphs or interpreting data.
Examples (graphs created by me, with the aid of this website and Excel):
Graph A: Constant Speed
The object portrayed in this graph starts at zero seconds, zero meters. Once in motion, it goes five meters per one second--meaning that it's slope is 5 meters over 1 second. This also means that the speed of the object, which is the same as the slope, is 5 meters per second.
The difference between the plotted points is consistently 5 over 1, meaning that the slope and therefore the speed of the object is constant. Its shape is due to the fact that the object moves the same number of meters each second as time goes by.
Graph B: Increasing Speed
In this graph, the object in motion increases by changing amounts. From 0 to 1 seconds, the object moves 0.5 meter. From 1 to 2 seconds, the object moves 1 meters, and from 2 to 3 seconds, the object moves 1.5 meters. The distance between the rest of the points increases by 0.5 meters for each second. From this we know that the slope is not staying the same over time, but increasing, meaning that the speed of the object is increasing over time as well. The shape of the graph is due to the fact that the object is moving more meters per second as time goes by.
Graph C: Decreasing Speed
In this graph, the difference between 1 and 2 seconds is 4 meters. The difference between 2 and 3 seconds is 3.5 meters, and the difference between 3 and 4 seconds is 3 meters. The difference between the points decreases by 0.5 meters each seconds. From this we can see that the slope is decreasing as time passes, meaning that the speed is decreasing as well. The shape of the graph is due to the fact that as time goes by, the object moves fewer meters each second.
Another method of visually representing motion is through a v-t, or velocity-time, graph.
Velocity is the rate of change of position of an object, or its change in speed over time.
This graph reflects a v-t version of the x-t graph above of constant speed. The x axis is time in seconds. The y axis is meters per second, which, on this graph, remains consistently at 5 meters per second.
On an x-t graph, a horizontal line of data would suggest that no change in distance occurred, and that therefore no motion occurred and that there was no speed. On a v-t graph, this means that the object's velocity did not change, but stayed constant, which means that the speed is increasing at a constant rate. Through this we see that similar shapes does not always mean similar information on different types of graphs.
This v-t graph represents the change in velocity for the data recorded in the x-t graph of increasing speed shown above. The x-axis is seconds, and the y axis is meters per second. Instead of curving as the x-t graph does, this graph is in a diagonal line that does not curve because the velocity is increasing at a constant rate.
On an x-t graph, data in this shape would suggest that the object was moving at a constant speed, but on a v-t graph, the data shows that the velocity of the object was increasing at a constant rate. Despite the differences in their shapes, both graphs accurately portray the same object's motion.
Note to Mr. Todd: Some of the graphs are from a graph-making website, but it got all gross and glitchy, so some of them are represented by Excel documents.
Relative Motion
We had a class assignment to create a video portraying relative motion and think of some "deep" questions to ask the class upon presentation.
This is our video:
Brought to you by Andrea Tate, Elly Bryant, Alley Bowers and myself.
The questions are as follows:
1. What motion do you see?
2. How does the motion differ from the person in the background to the two spinning girls?
3. Describe the motion of the spinning girls in relation to each other and in relation to the creeper in the background.
After viewing and discussing the situation with the class, it was decided that:
1. Two of the girls are spinning in a single circle, while one is walking in a straight line, getting closer to them for the first portion of the video, and then passing them and getting continually farther away.
2. Their motion differs in that the spinning girls travel in the same path for the duration of the video, the walking girl's path takes her forward onto new ground.
3. Relative to each other, the spinning girls are not moving because the distance between them is not changing as they spin. The girl in the background, however, gets closer and then farther away to the spinners, changing the distance between them and therefore moving relative to them.
This Thing You Call "Relative Motion"
So we did the project and whatnot, but what does it mean? And why does it matter?
Well, relative motion is defined as the motion of an object relative to another object.
For motion to occur, there must be something to compare the motion to--otherwise, how could you possibly know that it was moving? Think about how we describe a moving object--it got closer to me. It went away from the bus. For motion to occur, there has to be a change in distance, yes? And to measure this distance, we need a zero. This is the object/point in space that the object in motion is moving relative to. On a graph, it is the zero point (aptly named, this zero).
For example:
There are two cars parked next to each other in a parking lot, one red and one white. The red car starts up and drives forwards, away from the white car. Relative to the white car, the red car is moving, because the distance between them is changing. In this scenario, the white car is the zero and the red car is the object in motion.
But wait. If the definition of motion is a change in distance, wouldn't the white car be moving relative to the red car, too, even if it isn't the one driving away?
Yes. Exactly.
If you were to use the red car as the zero and the white car as the object in motion, the white car would be moving relative to the red car. In every situation, if object a is moving relative to object b, then the opposite is true as well. It just depends on where you place your zero.
So, keeping these ideas in mind, let's set up another scenario.
Now the red car and the white car are both driving down the road, the white one behind the red one, both going 35 miles per hour.
Because of their identical speed, the distance between them is not changing--but they're both being driven. Are they moving, relative to each other?
No.
The same rule still applies. If there is no change in distance between them, then, relative to each other, neither car was moving.
HOWEVER. This does not mean that neither car was moving--relative to other things.
Think of all the things that the cars' distances are increasing or decreasing from--the point in the parking lot at which they began driving, a sign that they pass by on the road, a hitchhiker left in the dust. So many possibilities.
Bottom line: If the distance is changing between objects, then they are both moving relative to each other.
And...why does it matter? Because all motion is relative. Without an object to compare it to (a zero), it is impossible describe (much less perceive) the motion of another object. Graphs cannot be made. Measurements cannot be taken. And the nice thing is, if you've ever described the motion of anything, you've been using the idea of relative motion your whole life without even realizing it.
How does it affect science?
To quote from this website:
"Motion means continuous change of position of an object with respect to an observer. To another observer in a different frame of reference the object may not be moving at all, or it may be moving in an entirely different manner. The motions of the planets were found in ancient times to appear quite complicated...By transferring to the frame of reference of an imaginary observer on the Sun, Johannes Kepler showed that the relative motion of the planets could be simply described in terms of elliptical orbits. The validity of one description is no greater than the other, but the latter description is far more convenient."
Basically, it helps us to better understand the motion of objects by being able to use different reference points to perceive it.
Parallax
I am looking at a flower vase at the end of the next room, perhaps fifteen or twenty feet away.
It appears to be right in the middle of my line of vision.
Viewing with left eye: The vase seems more left-ish.
Viewing with right eye: The vase seems much more to the right.
Switching back and forth between using one eye or the other, but never both at one time, the vase appears to be jumping from left to right and right to left, depending on the eye. But the vase isn't actually moving, is it?
This phenomena can be described as parallax, "the apparent displacement of an observed object due to a change in the position of the observer." (link)
Think about it. You aren't a cyclops. Your eyes are positioned a small space away from each other. When observing with both eyes, we see an object in a single place, but when we switch between using one eye or the other, the apparent difference in the position of the object shows that each eye percieves it from a different angle.
On a larger scale, you could stand to the right of a chair. The chair, obviously, appears to be at your left.
Move behind the chair and it appears to be directly in front of you.
Move to the left of the chair, and now it is on your right.
Just like before, the observed object has not changed its position, but the observer has.
However, distance makes a difference as to how you perceive an object. If I do the same thing that I did to the vase with the house outside of the window, all the way across the street, it doesn't seem to "move" at all. Why? My eyes are still perceiving the house from different points, right?
Fact is, the farther away an object is, the less its position will seem to change. Relative to the distance between my eyes and the house, the distance between my eyes is very small. To notice any change in postion, I would have to view the house from two points that are farther apart, like from one end of the block to the other. Likewise, the greater the distance is between the points of observation, the more the position of the observed object will appear to have changed.
So, when observing an object close to you that is aligned with a distant object and observing it from each individual eye, the close object will appear to move a lot in comparison to the distant object. They will no longer be aligned. This fact is particularly useful to astronomers.
How is parallax used?
Astronomers can use parallax to determine the distance to stars. It might seem like a far stretch from looking at a vase in your living room, but the principal is the same.
(Info and image pertaining to astronomy obtained from this website. Anything in quotations is text taken directly from the site.)
In this case, a person's eyes are not far enough apart to experience parallax when looking at a star. A much greater distance is needed--like the diameter of the earth's path around the sun, as shown at left.
"The parallax angle, P, is measured by comparing the nearby star's position to the stable position of distant background stars."
Basically, the astronomers use the known characteristics of triangles to determine the angles and therefore distances of the triangles created by the points of observation, the star being observed, and the distant stars being used as background reference points.
Using the operations of sine, cosign or tangent, if one of the other angles of the triangle is known (like the distance from the earth to the sun) and the angles have been found, the distance from the earth to the star is able to be calculated.
The Intersection
THE SITUATION:
One way to test one's knowledge of velocity is to apply it to a situation.
Let's say we have an intersection. Car A is stopped at a stop sign at the intersection's edge, waiting for its chance to go. The cross-traffic is traveling on the street perpendicular to Car A at a constant speed of 3 meters per second. All cars are 4.6 meters long.
The distance across the intersection, from curb to curb, is 14.23 meters.
The distance from the stop sign to the opposite curb is 19.4 meters.
The closest car in the cross traffic, Car B, is right with its front at the edge of the intersection and is going 3 meters per second. Car A is positioned with its front right at the stop sign. It is stopped, but has an acceleration of 2.237 meters per second per second.
Starting the cars at these positions, who will get across the intersection first? How long does it take each vehicle?
THE FORMULA:
These formulas can be used to figure the time an object takes to cover a certain area, change in position and velocity being known.
change in distance=(1/2 acceleration x time squared) + (initial velocity x time)
or change in X=1/2at^2 + Vit
also:
Change in distance=1/2 velocity x time
or change in X=1/2V x t
THE STEPS:
Taking the information given, these immediate conclusions can be made:
- Car A will have 24 meters to travel (length of car+distance from stop sign to intersection, 4.6+19.4).
-Car B will have 18.83 meters to travel (length of car+distance form one side of the intersection to the other, 4.6+14.23)
The distance of the car must be taken into account because the car is not fully out of the intersection until its end has passed the curb on the opposite side.
Car A:
change in X= 24
acceleration= 2.237
initial velocity= 0
Time is the variable that needs to be found.
24=1/2 2.237t^2 + 0t
24=1.1185t^2 + 0
21.457=t^2
t=4.632
The time it takes Car A to reach the other side of the intersection from the stop sign is 4.632 seconds.
Car B:
change in X= 18.83
velocity= 3
Here, again time needs to be found.
18.83=1/2 3 x t
18.83 = 1.5 x t
12.553 = t
The time it takes Car B to cross from one side of the intersection to the other is 12.553 seconds.
THE ANSWER:
Car A will reach the other side of the intersection before Car B, even though Car A has a larger distance to cover. This is due to its acceleration.
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