183_notes:displacement_and_velocity

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183_notes:displacement_and_velocity [2017/08/24 20:14] – [Lecture Video] pwirving183_notes:displacement_and_velocity [2021/02/18 21:17] (current) – [Velocity and Speed] stumptyl
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 +Section 1.4 and 1.6 in Matter and Interactions (4th edition)
 +
 ===== Constant Velocity Motion ===== ===== Constant Velocity Motion =====
  
-**Our job in mechanics is to predict or explain motion. So, all the models and tools that we develop are aimed at achieving this goal.**+//Our job in mechanics is to predict or explain 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).+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).**
  
 ==== Lecture Video ==== ==== Lecture Video ====
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 ==== Motion (Changes of Position) ==== ==== Motion (Changes of Position) ====
  
-//Displacement is a vector quantity that describes a change in position.//+**Displacement** is a vector quantity that describes a change in position.
  
-{{ course_planning:course_notes:displacement.png|Displacement vector}}+{{ week1_constantv.png|Displacement vector}}
 The displacement vector ($\Delta \vec{r}$) describes the change of an object's position in space (i.e., a change in location). So, you can think of the displacement vector as //displacement = change in position = final location - initial location//. This change in position is represented in the diagram to the right. Mathematically, we represent the displacement like this: The displacement vector ($\Delta \vec{r}$) describes the change of an object's position in space (i.e., a change in location). So, you can think of the displacement vector as //displacement = change in position = final location - initial location//. This change in position is represented in the diagram to the right. Mathematically, we represent the displacement like this:
  
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 where the subscripts "f" and "i" describe the //final// and //initial// locations respectively. You will often find it useful to have subscripts like those used above to distinguish between similar quantities (e.g., positions) that occur at different times (e.g., before some motion occurs and after that motion occurs). where the subscripts "f" and "i" describe the //final// and //initial// locations respectively. You will often find it useful to have subscripts like those used above to distinguish between similar quantities (e.g., positions) that occur at different times (e.g., before some motion occurs and after that motion occurs).
  
-In one dimension, you might consider motion along a specific coordinate axis or, if you like, the number line. In that case, you can still talk about displacement "//in the x-direction//." Mathematically, we represent that kind of displacement like this:+In one dimension, you might consider motion along a specific coordinate axis or, if you like, the number line. In that case, you can still talk about displacement "in the x-direction." Mathematically, we represent that kind of displacement like this:
  
 $$\Delta x = x_f - x_i$$ $$\Delta x = x_f - x_i$$
  
-**Note that this displacement maybe positive, negative, or zero, as this is the component of the displacement vector in the x-direction.**+//Note that this displacement maybe positive, negative, or zero, as this is the component of the displacement vector in the x-direction.//
  
-The units of displacement are units of length, which are typically the SI units of meters (m).+The units of displacement are units of length, which are typically the SI units of **meters (m).**
  
 /* Add a little about distance versus displacement */ /* Add a little about distance versus displacement */
-==== Velocity and Speed ====+===== Velocity and Speed =====
  
-//Velocity is a vector quantity that describes the rate of change of the displacement.//+**Velocity** is 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 ==== 
 + 
 +**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:
  
 $$\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t} = \dfrac{\vec{r}_f - \vec{r}_i}{t_f - t_i}$$ $$\vec{v}_{avg} = \dfrac{\Delta \vec{r}}{\Delta t} = \dfrac{\vec{r}_f - \vec{r}_i}{t_f - t_i}$$
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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 ===+==== 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 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?
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 $$v_{x,avg} = \dfrac{\Delta x}{\Delta t} \approx \dfrac{v_{ix} + v_{fx}}{2}$$ $$v_{x,avg} = \dfrac{\Delta x}{\Delta t} \approx \dfrac{v_{ix} + v_{fx}}{2}$$
  
-This equation only hold exactly if the velocity changes linearly with time ([[183_notes:constantf|constant force motion]]). It might be a very poor approximation if velocity changes in other ways.+//__This equation only hold exactly if the velocity changes linearly with time ([[183_notes:constantf|constant force motion]])__.// It might be a very poor approximation if velocity changes in other ways. 
 + 
 +\\
  
-=== Instantaneous Velocity ===+==== Instantaneous Velocity ====
  
-Instantaneous velocity ($\vec{v}$) describes how quickly an object is moving at 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 the instantaneous velocity like this:+**Instantaneous velocity** ($\vec{v}$) describes how quickly an object is moving at 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 the instantaneous velocity like this:
  
 $$\vec{v} = \lim_{\Delta t \rightarrow 0} \dfrac{\Delta \vec{r}}{\Delta t} = \dfrac{d\vec{r}}{dt}$$ $$\vec{v} = \lim_{\Delta t \rightarrow 0} \dfrac{\Delta \vec{r}}{\Delta t} = \dfrac{d\vec{r}}{dt}$$
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 $$v_x = \lim_{\Delta t \rightarrow 0} \dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}$$ $$v_x = \lim_{\Delta t \rightarrow 0} \dfrac{\Delta x}{\Delta t} = \dfrac{dx}{dt}$$
 +\\
  
-=== Speed ===+==== Speed ====
  
-//Speed is a scalar quantity that describes that distance (not the displacement) traveled over an elapsed time.//+**Speed** is a scalar quantity that describes that distance (not the displacement) traveled over an elapsed time.
  
-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:
  
 $$s =\dfrac{d}{t}$$ $$s =\dfrac{d}{t}$$
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 $$|\vec{v}| = \sqrt{v_x^2+v_y^2+v_z^2}$$ $$|\vec{v}| = \sqrt{v_x^2+v_y^2+v_z^2}$$
  
-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 ==== ==== Predicting the motion of objects ====
  
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 Constant velocity motion is motion that occurs when an object travels in a straight line at constant speed, or, more realistically, can be considered to //approximately// travel in a straight line at constant speed. That is, for this object its velocity remains unchanged because there are [[183_notes:momentum_principle|no external influences on the motion]]. Constant velocity motion is motion that occurs when an object travels in a straight line at constant speed, or, more realistically, can be considered to //approximately// travel in a straight line at constant speed. That is, for this object its velocity remains unchanged because there are [[183_notes:momentum_principle|no external influences on the motion]].
  
-For constant velocity motion, the velocity is a constant vector and, hence, the average and instantaneous velocities are equivalent. That is, **for constant velocity motion only**:+For constant velocity motion, the velocity is a constant vector and, hence, the average and instantaneous velocities are equivalent. That is, //for constant velocity motion only//:
  
 $$\vec{v}_{avg} = \vec{v}$$ $$\vec{v}_{avg} = \vec{v}$$
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