course_planning:course_notes:constantv

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course_planning:course_notes:constantv [2014/06/19 01:47] – caballerocourse_planning:course_notes:constantv [2014/07/08 13:20] (current) – [Speed and Velocity] caballero
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-===== The Constant Velocity Model =====+===== Constant Velocity Motion =====
  
-The simplest model of motion is for an object that moves in a straight line at constant speed. You can use this simple model to build your understanding about the basic ideas of motion, and the different ways in which you will represent that motion.+Our job in mechanics is to predict motion. So, all the models and tools that we develop are aimed at achieving this goal. 
 + 
 +The simplest model of motion is for an object that moves in a straight line at constant speed. You can use this simple model to build your understanding about the basic ideas of motion, and the different ways in which you will represent that motion. At the end of these notes, you will find the position update formula, which is a useful tool for predicting motion (particularly, when it comes to constant velocity motion).
  
 ==== Motion (Changes of Position) ==== ==== Motion (Changes of Position) ====
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 **Displacement:** A vector quantity that describes a change in position. **Displacement:** A vector quantity that describes a change in position.
  
-The displacement vector ($\Delta \vec{r}$) describes the change of an object's position in space. The displacement vector describes a change in location. So, you can think of it as //displacement = change in position = final location - initial location//. This is represented in the diagram to the right **(NEED DISPLACEMENT FIGURE)**. Mathematically, we represent the displacement like this:+{{ course_planning:course_notes:displacement.png|Displacement vector}} 
 +The displacement vector ($\Delta \vec{r}$) describes the change of an object's position in space. The displacement vector describes a change in location. So, you can think of it as //displacement = change in position = final location - initial location//. This is represented in the diagram to the right. Mathematically, we represent the displacement like this:
  
 $$\Delta \vec{r} = \vec{r}_{final}-\vec{r}_{initial}-\vec{r}_f - \vec{r}_i$$ $$\Delta \vec{r} = \vec{r}_{final}-\vec{r}_{initial}-\vec{r}_f - \vec{r}_i$$
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 **Velocity:** A vector quantity that describes the rate of change of the displacement. **Velocity:** A vector quantity that describes the rate of change of the displacement.
 +
 +=== Average Velocity ===
  
 Average Velocity ($\vec{v}_{avg}$) describes how an object changes its displacement in a given time. To compute an object's average velocity, you will need the position of the object at two different times. You can think of it as //average velocity = displacement divided by time elapsed//. Mathematically, we can represent the average velocity like this: Average Velocity ($\vec{v}_{avg}$) describes how an object changes its displacement in a given time. To compute an object's average velocity, you will need the position of the object at two different times. You can think of it as //average velocity = displacement divided by time elapsed//. Mathematically, we can represent the average velocity like this:
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 where $t_f - t_i$ is always positive, but $x_f-x_i$ can be positive, negative, or zero because it represents the displacement in the x-direction, which is a vector component. where $t_f - t_i$ is always positive, but $x_f-x_i$ can be positive, negative, or zero because it represents the displacement in the x-direction, which is a vector component.
 +
 +
 +=== Approximate Average Velocity ===
 +
 +The average velocity is defined as the displacement over a given time, but what about the //arithmetic// average velocity? How do the arithmetic average velocity and average velocity compare?
 +
 +The arithmetic average velocity is a approximation to the average velocity.
 +
 +$$v_{x,avg} = \dfrac{\Delta x}{\Delta t} \approx \dfrac{v_{ix} + v_{fx}}{2}$$
 +
 +This equation only hold exactly if the velocity changes linearly with time (constant force motion). It might be a very poor approximation if velocity changes in other ways.
 +
 +=== Instantaneous Velocity ===
  
 Instantaneous velocity ($\vec{v}$) describes how quickly an object is moving a specific point in time. If you consider the displacement over shorter and shorter $\Delta t$'s, your computation will give a reasonable approximation for the instantaneous velocity. In the limit that $\Delta t$ goes to zero, your computation would be exact. Mathematically, we represent this instantaneous velocity like this: Instantaneous velocity ($\vec{v}$) describes how quickly an object is moving a specific point in time. If you consider the displacement over shorter and shorter $\Delta t$'s, your computation will give a reasonable approximation for the instantaneous velocity. In the limit that $\Delta t$ goes to zero, your computation would be exact. Mathematically, we represent this instantaneous velocity like this:
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 **Speed:** A scalar quantity that describes that distance (not the displacement) traveled over an elapsed time. **Speed:** A scalar quantity that describes that distance (not the displacement) traveled over an elapsed time.
 +
 +
 +=== Average speed ===
  
 Average speed ($s$) describes how quickly an object covers a given distance in a given amount of time. So, you can think of it as //average speed = total distance traveled divided by total time elapsed//. Mathematically, we represent the average speed like this: Average speed ($s$) describes how quickly an object covers a given distance in a given amount of time. So, you can think of it as //average speed = total distance traveled divided by total time elapsed//. Mathematically, we represent the average speed like this:
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 Notice that the instantaneous velocity is equivalent to the magnitude of the velocity vector and, therefore, is a positive scalar quantity. Notice that the instantaneous velocity is equivalent to the magnitude of the velocity vector and, therefore, is a positive scalar quantity.
 +==== Predicting the motion of objects ====
 +
 +We can rewrite the definition of average velocity above to give us information about the displacement of an object,
 +
 +$$ \Delta \vec{r} = \vec{r}_f - \vec{r}_i = \vec{v}_{avg} \Delta t$$
 +
 +This equation tells us that given a certain average velocity ($\vec{v}_{avg}$) over a known time interval ($\Delta t$), an object will experience a particular displacement ($\Delta \vec{r}$). By moving the initial position over to the left side, we get the "position update" formula, 
 +
 +$$ \vec{r}_f = \vec{r}_i + \vec{v}_{avg} \Delta t $$
 +
 +which allows us to predict the location of an object given its initial position and average motion. This formula is a very powerful because it allows us to predict where an object will be given only information about it now.
  
 ==== What's so special about constant velocity motion? ==== ==== What's so special about constant velocity motion? ====
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 $$\vec{v}_{avg} = \vec{v}$$ $$\vec{v}_{avg} = \vec{v}$$
  
-The object changes its position at a constant rate. +The object changes its position at a constant rate. In the position update formula, we can replace the average velocity with simply the instantaneous velocity,
  
-==== Representations of Constant Velocity Motion ====+$$ \vec{r}_f = \vec{r}_i + \vec{v} \Delta t $$
  
-=== Graphs ===+==== Pre-Lecture - Displacement and Velocity ====
  
-=== Diagrams ===+{{youtube>J9fpQNVGIdg?large}}
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