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In the previous module, we developed a mathematical description of position. In this module, we develop our first mathematical model of motion: average velocity and instantaneous velocity.
After working through this module, you should be able to:
- Define Average Velocity.
- Give a mathematical definition of instantaneous velocity in terms of position and time.
- Explain the difference between velocity and speed.
- Describe the conditions for the application of the Constant Velocity Model of motion and know its Law of Change.
- Recognize or create a position versus time graph for an object moving with constant velocity.
- Given the position versus time graph for an object moving with constant velocity, determine the object's velocity.
Illustrative Example: Return to Campus
Consider again the example from the previous module:
A student rushes from their dorm room to the physics building in 2 minutes. After spending 1 minute turning in their homework, the student runs to the cafeteria in 2 minutes. The student eats lunch for 12 minutes, then walks to the library in 6 minutes.
Further, recall the coordinate system and positions we assigned to the locations, with the positive x-axis directed east:
Now, suppose you were asked to determine the velocity of the student during their rush from dorm to physics building as described in the problem statement. All you know is the total displacement (2 blocks East) and the duration of their rush (2 minutes). In this case we can only determine their average velocity over the time interval [0, 2] (this notation means every point from t=0 to t=2 inclusivee of the endpoints).
Average Speed - a Useful Concept
Suppose that, as above, you know only that an object starts from xi at t = ti and has moved to x(t) at time t. In many cases exact details of this motion don’t matter. In this example there may be a hill in the intervening block, but all we’re really interested in is that it took 2 minutes to go two blocks. Under these circumstances, we’re interested in the ‘’’average speed’’’. This is defined as
, and may be thought of as
The notation vavg[t,ti] emphasizes that the average velocity is a function of the particular time interval over which the motion occurs rather than any particular instant of time. In this case the average velocity is + 1 block per minute. The "+" sign indicates that the displacement is to the East or positive x.
In many situations, it is quite sufficient to use the concept of average velocity. In the case above, it seems quite reasonable to assume that the student maintains a constant pace during each segment of their trip (though that pace may be different in different segments). Modeling a journey by breaking it into segments and then using the assumption of a (different) constant speed within the various parts is a very useful approximation that is extremely common in trip planning.
Another justification for motion with constant velocity comes from Newton's First Law stating that the natural state of motion for objects experiencing no interactions is motion with constant velocity. This justifies the importance of the Constant Velocity (Zero Acceleration) as our first Model of motion in this course. It will be discussed more fully and in the context of our hierarchy of models in the next unit.
Note that when an object is moving at constant velocity, the object moves along a straight line (1 - dimensional motion). The velocity is a vector, therefore when we say that the velocity is constant we are implying constant magnitude and constant direction. In the above mathematical representation of the position as a function of time we used a coordinate system with the x - axis parallel to the object's motion, but did not bother to indicate this choice by using x-subscripts. When dealing with an object such as a ball moving at constant velocity horizontally but with gravity acting in the vertical direction, we’d certainly use x-subscripts for the constant velocity motion.
We can now illustrate the application of the Motion with Constant Velocity Model by finding the velocity of the student mentioned in the example given at the top of this page during each leg of the trip. We have already set up a coordinate system for this problem, so we know how to describe the position of the student with respect to the zero in this coordinate system. We have not yet decided how to report the times in this example, however. Suppose that we define our time coordinates so that the example problem begins at time t = 0. Then we have all the information we need to find the student's velocity during each leg of their trip under the assumption of constant velocity during each leg. The answers for each leg are available below, but initially hidden to give you a chance to work them out yourself.
|Velocity Calculation for Leg 1 (Dorm Room to Physics Building)|
|In the first leg of the trip described in the example, the student leaves their dorm room and rushes to the physics building in 2 minutes. Based upon the coordinate system we defined and our choice to define t = 0 at the beginning of the trip, we have the initial conditions: |
This leg of the trip ends at the final location and time:
Thus, under the assumption that the student's velocity is constant, we find it to be:
Note that the units of velocity are distance/time.
|Velocity Calculation for Leg 2 (Turning in Homework)|
| After reaching the physics building, the student spends four minutes in the physics building turning in homework. Clearly, the velocity for this portion of the trip will be zero. Thus, we model it separately of the trip to the physics building. The initial conditions for this leg are the same as the final conditions for the previous portion of the trip: |
The final position and time are:
Thus, the Constant Velocity Model gives:
which is the result we expected.
|Velocity Calculation for Leg 3 (Physics Building to Cafeteria)|
| After turning in their homework, the student hurries to the cafeteria in 2 minutes. Again, the initial conditions are equal to the conditions at the end of the previous leg: |
The final conditions are:
This velocity is negative, which, in our coordinate system (as defined in the previous module), implies that the velocity is in the opposite direction of the velocity in the first leg of the trip. Specifically, if the +x-axis points east, the first leg gave an eastward velocity and this 3rd leg represents a westward velocity. Note: the sign of this velocity is because velocity is a vector, and due to the definition of it in terms of the displacement vector, the velocity vector will point in the direction of the displacement. This is true for both the average velocity and the instantaneous velocity..
|Velocity Answers for Remaining Legs|
|Using the same techniques employed in the previous legs, you can show that the student's velocity during the time that they spend at the cafeteria is zero, and their velocity during the walk to the library is positive 0.67 blocks/minute. The direction of this velocity vector is positive, meaning the same direction as the veolcity of the first leg of the student's journey.|
Graphical Representation of the Model
Looking at the mathematical representation of the velocity model after our rearrangement, we can see that the expression for velocity:
is equivalent to the algebraic expression for the slope of a position versus time graph. Thus, if the velocity is constant for a portion of a given motion, we expect that the position versus time graph will be linear over that time interval with a constant slope equal to the velocity. Thus, a position versus time graph is a simple and informative graphical representation of the Constant Velocity (Zero Acceleration) Model. The position versus time graph for the example we have been discussing would be:
If velocity is constant, it is technically always possible to align the x-axis of your coordinate system with the direction of the velocity. Sometimes, however, this is not a desirable thing to do. If the velocity points in a direction that is not perfectly aligned with the x-axis, you must use vector components to analyze the motion. Essentially, we add another copy of the model for each direction that is needed. Thus, for two dimensional motion, we would use two equations simultaneously:
The total velocity would then be found by combining the x and y components of the velocity using the standard rules for vector components.
Speed, Average Velocity, and Average Speed
Three terms that relate to velocity and are commonly encountered in mechanics problems are:
- Speed is the magnitude of the velocity vector. Speed is always positive and is not a vector.
- The average velocity for a given trip is the velocity that would be needed to complete the entire trip at constant velocity. In other words, it is the total displacement for the trip divided by the total time.
- The average speed is the total distance covered in the trip divided by the total time.
Let's say you wanted to describe the student's motion more realistically and in more detail. If you were a student at the same school, you might know that there is a hill between the dorm and the physics building; therefore the student certainly moved with less than the average velocity on the first leg up the hill and faster down hill. THerefore breaking the dorm --> physics building leg into two smaller segments, each with its own average velocity, would certainly provide a better understanding of the student's motion than does the single average velocity used above.
If we really want to describe the motion of the student most accurately, we could subdivide the journey into an infinite number of segments, each with its own average velocity. This procedure leads to the concept of instantaneous velocity, which is a function defined at each instant of time t of the journey. It may be defined as:
This relationship emphasizes that the official definition of instantaneous velocity is simply the time derivative of position, and that this is the same as the average velocity over shorter and shorter time intervals around time t. Generally we leave out the word "instantaneous" and simply refer to the above concept as "velocity"
Velocity is the simplest and defining measure of the translational motion of a particle.
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