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course_planning:course_notes:constantv [2014/06/19 01:47] – caballero | course_planning:course_notes:constantv [2014/07/08 13:20] (current) – [Speed and Velocity] caballero | ||
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- | ===== The Constant Velocity | + | ===== Constant Velocity |
- | 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. |
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+ | 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, | ||
==== Motion (Changes of Position) ==== | ==== Motion (Changes of Position) ==== | ||
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**Displacement: | **Displacement: | ||
- | The displacement vector (Δ→r) describes the change of an object' | + | {{ course_planning: |
+ | The displacement vector (Δ→r) describes the change of an object' | ||
Δ→r=→rfinal−→rinitial−→rf−→ri | Δ→r=→rfinal−→rinitial−→rf−→ri | ||
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**Velocity: | **Velocity: | ||
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+ | === Average Velocity === | ||
Average Velocity (→vavg) describes how an object changes its displacement in a given time. To compute an object' | Average Velocity (→vavg) describes how an object changes its displacement in a given time. To compute an object' | ||
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where tf−ti is always positive, but xf−xi can be positive, negative, or zero because it represents the displacement in the x-direction, | where tf−ti is always positive, but xf−xi can be positive, negative, or zero because it represents the displacement in the x-direction, | ||
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+ | === Approximate Average Velocity === | ||
+ | |||
+ | The average velocity is defined as the displacement over a given time, but what about the // | ||
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+ | The arithmetic average velocity is a approximation to the average velocity. | ||
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+ | vx,avg=ΔxΔt≈vix+vfx2 | ||
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+ | 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 (→v) describes how quickly an object is moving a specific point in time. If you consider the displacement over shorter and shorter Δt's, your computation will give a reasonable approximation for the instantaneous velocity. In the limit that Δt goes to zero, your computation would be exact. Mathematically, | Instantaneous velocity (→v) describes how quickly an object is moving a specific point in time. If you consider the displacement over shorter and shorter Δt's, your computation will give a reasonable approximation for the instantaneous velocity. In the limit that Δt goes to zero, your computation would be exact. Mathematically, | ||
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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. | ||
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+ | === 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, | 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, | ||
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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 ==== | ||
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+ | We can rewrite the definition of average velocity above to give us information about the displacement of an object, | ||
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+ | Δ→r=→rf−→ri=→vavgΔt | ||
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+ | This equation tells us that given a certain average velocity (→vavg) over a known time interval (Δt), an object will experience a particular displacement (Δ→r). By moving the initial position over to the left side, we get the " | ||
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+ | →rf=→ri+→vavgΔt | ||
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+ | 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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→vavg=→v | →vavg=→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 |
- | === Graphs | + | ==== Pre-Lecture - Displacement and Velocity ==== |
- | === Diagrams === | + | {{youtube> |