There are two different studies of motion. The first area focuses on describing and predicting the motion. It is called kinematics. The second area, called dynamics, goes one step deeper. This adds the dimension of describing why the motion occurs, e. g., the concept of force.
There are many ways to describe motion. We will focus on the following ways: written description, motion maps (strobe photographs), graphical representation, and mathematical equations. We will develop these concepts by reviewing some special cases or paradigms.
What is needed to describe motion? Every day situations use terms like start, finish, position, location, distance, speed, velocity, and acceleration. In addition to the terms, we can fix numbers to the terms. The numbers are related to some defined number line, called a coordinate axis. The numbers are also accompanied by some reference term called a unit.
The coordinate axis is a way of quantifying information. We use the concept every day. For example, time could be thought as a coordinate axis. Often you will find historical events on a "time line", i. e., a coordinate axis. We will use this, realizing that we can only go back in time in our imagination. The other coordinate axis we will use is a number line fixed in space. This will be used to quantify our descriptions of motion. The coordinate axii will be numbered in a regular pattern, but the size of the step and the location of zero is up to the person to define.
The coordinate axis step size is identified by the unit. The units allow others to scale the numbers. For example, "I am going 25 miles per hour" conveys a meaning to anyone familiar with this reference. The unit, "miles per hour", conveys a basic quantity or size which others will understand. The number "25" how many of those basic quantities are present. It is important to use standard units to convey the meaning of the number to others. It is also important to always state the unit.
An Aside on Units: The Mars orbiter loss provides an excellent example of unit communication. Numbers were provided with assumed units. Unfortunately, the people involved were using different units (basic quantities for the coordinate axii). As a result, calculations were off and the orbiter was lost. Similarly, imagine you were told "I'll pay you 5 an hour". You assumed the person meant the standard unit of dollars, but the person always spoke in terms of cents. This miscommunication would cost you dearly. To expand this, the person could have said "5 zalladans an hour". Even though the person stated the units, you would not know how to interpret the number since the unit is nonstandard. The unit is important to complete the meaning. It must be something that everyone involved understands, i. e., has a personal reference to what the size means. (This is one problem the US has with the metric system. The system is a standard, but most people don't have a working understanding of the quantity the units represent.) And, the unit should be stated, not assumed.
Constant Velocity Paradigm
Lets begin to look at motion by imagining something stationary.
We could say "it is at 0 meters". This identifies what we call the "position" or "location". The number and unit refer to the chosen coordinate axis. To express this mathematically, we would use the equation x(t) = 0 m (position as a function of time equals zero meters). We could say "it is stopped". This identifies how the position is changing as time progresses. We use the term velocity to describe this. Mathematically, it would be v = 0 m/s. (In everyday situations we also use the term "speed".)
To broaden this example, let's have the object move. Specifically, let the object move in a uniform way. That is, we only perceive a change in position, nothing else. This motion is called constant velocity. Two examples are rolling a ball across a horizontal table or watching a car move along a straight road with the same number on the speedometer.
We could imagine taking a strobe light photograph on one piece of film. Each flash would show where the object was at a given moment in time as defined by our clock reading. Alternately, imagine taking a movie of object keeping the camera stationary. If each frame of film was overlaid, you would see where the object was at different clock readings ("times"). Such a picture is shown below. It is called a motion map. (Note, the starting time and location is identified, this is necessary. Identifying the time interval is not needed, but the time interval must be constant.)
If we were to combine the position coordinate axis with a time coordinate axis, we get the position versus time graph shown below.
If we were to look at the same motion using a strobe time interval of 0.5 seconds, we get the position versus time graph shown below.
Continuing to decrease the strobe time interval would lead ultimately to a straight line.
The straight line is a graphical way to represent the motion. We will correlate certain graph characteristics with certain motion properties. If the object moved faster, how would the motion map and graph change? The dots would get further apart and the line would get steeper. If the object were to go slower, the dots would get closer and the line would get less steep. This is illustrated below.
To relate the motion to the graph we see the starting point (time is set to zero) corresponds to the y axis value. The slope of the line relates to how fast the object is moving. Calculating the change in position for one unit of time (1 second), yields the same number and units. This quantity is called the velocity. It is calculated as follows:
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For constant velocity motion, the plot is always a straight line. The slope of the line is the velocity. The stationary case is a special constant velocity situation where the velocity is zero. The order of the calculation is important. Look at the motion map and plots below as an illustration.
A written description of this motion could be like this. "Start at a position of 10 meters and move towards smaller position numbers at a rate of 2 meters smaller every second." Or, "Start at 10 meters and move with a velocity of -2 meters per second." On the motion map the object is moving toward zero (more negative numbers). On the position versus time plot, it is a straight line with a slope of - 2 m/s. For the velocity versus time plot, the velocity is constant (slope of zero) and has a value of -2 m/s.
This points out an important difference between velocity and the term speed. The negative velocity above means the object is moving toward smaller position numbers, toward negative infinity on the number line. A positive velocity means the object is moving towards larger position numbers, toward positive infinity. Speed only represents the amount of motion, i. e., the positive amount or absolute value of the velocity. If you were told to get on I 75 and travel at a speed of 55 miles per hour to get to a free chocolate festival, you would need to ask for clearer directions. How long do I travel at that speed? Speed and velocity do not tell you how far or where you are. They only tell you how where you are is changing. You would also need to know whether to head north or south. That is what the sign of the velocity gives you, the direction to move along the number line. Given just a speed is not enough. You need a direction and some idea of the change in position (how long to travel that fast or how far to travel).
The how far to travel is measured as where the object ends (position) minus where it starts. This is called the displacement and is represented by x2-x1 in the velocity equation. It also is a positive or negative number. The sign relates the direction the object moved compared to the starting point. A common term used everyday instead of displacement is distance. Distance is like speed. It relates how much the object moved, but does not give the direction. "Go on I 75 and travel a distance of 55 miles" is not enough information to get to the destination. A direction is needed. Displacement includes that direction.
As a summary of constant velocity. The motion map shows the object making equally sized changes in location. The position versus time plot is linear with the slope equaling the velocity. The velocity versus time plot is linear with a slope of zero.
You can use the sonic range finder to look at this. Plot position (distance) versus time. Ask a person how the graph would look when someone walks away from the range finder (the finder is a zero, away from it is going toward positive infinity). Have someone walk away from the finder collecting data. Ask how the plot would change if someone walked away faster or slower. Try it. Ask what would happen if someone started away from the range finder and walked toward it. Repeat the exercise using the velocity versus time plot instead. Try using the motion matching plots of position and velocity as well.
Constant Acceleration Paradigm
Now, let's imagine sequentially combining constant velocity motions. Imagine the velocity was constant for five seconds and then changed, constant for five seconds and changed, etc. Each five second interval would be a constant velocity case. The position versus time is linear for each five second segment, but the slope is different. If the time interval got smaller, the linear segment begin to blend together. Below are position and velocity plots for time intervals of 5, 2.5, and 1 seconds each.
If we were to continue this pattern, eventually the velocity steps would become smooth and linear. The position curve would become smooth and quadratic in time.
We can also identify a new quantity which relates how quickly the velocity is changing with the passage of time. This is called the acceleration. On a graph of velocity versus time, this change in velocity divided by the change in time is the same as the slope calculation. Thus, the relationship between velocity and acceleration is very much like the relationship between position and velocity. Acceleration is the slope of a velocity versus time graph. The acceleration relates how the velocity changes with time. The larger the magnitude of the acceleration, the more quickly velocity changes (the steeper the slope). The sign of acceleration relates to the direction the velocity is changing. Positive acceleration is increasing velocity (becoming more positive or toward positive infinity). Negative acceleration is decreasing velocity (becoming less positive or toward negative infinity). Acceleration does not tell the value of the velocity, only how it changes. Since velocity does not change in the constant velocity case, the acceleration is zero. We will restrict the discussion to the case where acceleration is constant. If is not constant, calculus needs to be employed. (When acceleration changes, we can look at its plot and its slope. The slope of acceleration, which relates how acceleration changes with passing time, is called the jerk.) Constant acceleration, in equation form, is:
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Below are the motion map, position, velocity, and acceleration versus time plots when the step size is taken to infinitesimally small. These plots relate a constant acceleration of +1 [m/s]/s with the associated velocity and position. Note the linear growth of the velocity and the quadratic growth of the position.
Since velocity is related to change in position and the time interval on a motion map is constant, the object on a motion map will get closer together or further apart as represented by the velocity for that interval. That is when the velocity magnitude (speed) is large, the separation is large, when it is small, the separation is small. (Remember, the sign of the velocity relates which direction to move on the number line only and nothing associated with velocity reveals what position the object is at. Velocity only reveals how the position is changing.)
To emphasize the similar relationship of velocity to acceleration compared to position to velocity, positive and negative acceleration cases are shown below. They follow the same pattern we found position to velocity for the constant velocity case. Some will use the term de-acceleration for negative acceleration. This term adds confusion. Typically, it is used to imply slowing to a stop this is too limiting. Negative acceleration simply means the velocity is changing towards more negative numbers. But, the sign is simply based on the choice of coordinate systems made at the beginning. For example, we typically say going up from the ground is positive. If a ball is dropped, it is going toward negative positions (negative displacement) and has negative velocity which has a magnitude that is getting larger and larger. Thus, it has an acceleration which is negative (direction is down) and an increasing speed. The term de-acceleration does not fit this situation.
Finally, three equations can be used to describe constant acceleration motion. These are in addition to the plots and motion maps described above.
Use the range finder to test these plots. Try the same exercises used for position and velocity. Typically, it is very easy to interpret the position versus time plots. The velocity and acceleration plots each add a new level of complexity. Give it a try.
Summary of Relationships
|
Relationships as time, t, changes |
Position, x |
Velocity, v |
Acceleration, a |
|
Position, x |
Tells the location relative to a chosen coordinate system |
Velocity tells how quickly and in which direction the position is changing in the defined coordinate system. When the velocity is constant the position changes uniformly with time, i. e., x vs t is linear. Velocity is the slope of x vs t. Velocity is the deriviative of x wrt t |
Accleration tells if the position is changing in a non-linear manner in the defined coordinate system. When the accleration is constant and non-zeron the position changes non-liniearly with time, i. e., v vs t is not linear. Accleration is the slope of the slope of x vs t. Accleration is the second dirivative of x wrt t. |
|
Velocity, v |
Velocity tells how quickly and in which direction the position is changing in the defined coordinate system. When the velocity is constant the position changes uniformly with time, i. e., x vs t is linear. Velocity is the slope of x vs t. Velocity is the deriviative of x wrt t |
Velocity tells how quickly and in which direction the position is changing in the defined coordinate system. |
Accleration tells how quickly and in which direction the velocity is changing in the defined coordinate system. When the accleration is constant the velocity changes uniformly with time, i. e., v vs t is linear. Accleration is the slope of v vs t. Accleration is the first deriviative of v wrt t. |
|
Acceleration, a |
Accleration tells if the position is changing in a non-linear manner in the defined coordinate system. When the accleration is constant and non-zeron the position changes non-liniearly with time, i. e., v vs t is not linear. Accleration is the slope of the slope of x vs t. Accleration is the second dirivative of x wrt t. |
Accleration tells how quickly and in which direction the velocity is changing in the defined coordinate system. When the accleration is constant the velocity changes uniformly with time, i. e., v vs t is linear. Accleration is the slope of v vs t. Accleration is the first deriviative of v wrt t. |
Accleration tells how quickly and in which direction the velocity is changing in the defined coordinate system. |
Changing Accleration
It is quite possible the accleration is not constant. The quantity which measure the change in accleration with passing time is called the jerk. In other words, the jerk is to accleration as accleration is to velocity and as velocity is to position. The level of complexity can increase very rapidly. But, we can often approximate situations just as we did earlier by connecting different constant velocity situations together. Thus, going further than the constatn acceleration paradigm is not necessary typically.
Multiple Dimensional Motion
All motion can be described after identifying a coordinate line it occurs along. Most motion is complicated by the fact it does not move along any one straight line. In these cases using multiple, perpendicular coordinate lines are used to describe the motion. The motion is then analyzed relative to the motion along each line. It is as if the other coordinate line were "squashed" down so that all the points were at one point. That is, each direction is analyzed as if the other direction didn't exist. For example, when something is moving through the air, we would look at the vertical motion and at the horizontal motion separately.
This is not easy to comprehend, but it is a general principle. Notice, the motion maps could be obtained by drawing a perpendicular line from each point to the axis representing the direction for the motion map. In other words, the complicated two dimensional motion can be discussed as two perpendicualr one dimensional cases. This is true in general, i. e., complex situations of motion can be broken down into the sum of many perpendicular one dimensional cases.