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183_notes:acceleration [2014/07/10 20:54] – [Acceleration] caballero | 183_notes:acceleration [2021/02/04 23:23] (current) – [Why not just use change in momentum?] stumptyl | ||
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==== Newton' | ==== Newton' | ||
- | The Momentum Principle (or Newton' | + | The Momentum Principle (or Newton' |
$$\vec{F}_{net} = m\:\vec{a} = \dfrac{\Delta\vec{p}}{\Delta t}$$ | $$\vec{F}_{net} = m\:\vec{a} = \dfrac{\Delta\vec{p}}{\Delta t}$$ | ||
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$$\vec{a} = \dfrac{\Delta\vec{p}}{m\: | $$\vec{a} = \dfrac{\Delta\vec{p}}{m\: | ||
- | where the last two equals signs hold only if the mass of the system is not changing. | + | __//where the last two equals signs hold only if the mass of the system is not changing. |
+ | //__ | ||
==== Acceleration ==== | ==== Acceleration ==== | ||
- | //Acceleration is a vector quantity that quantifies how quickly the velocity of a system is changing.// | + | **Acceleration** is a vector quantity that quantifies how quickly the velocity of a system is changing. |
The acceleration can be defined in two ways and each is useful in different problems or ways of thinking. From Newton' | The acceleration can be defined in two ways and each is useful in different problems or ways of thinking. From Newton' | ||
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$$\vec{a} = \dfrac{\vec{F}_{net}}{m}$$ | $$\vec{a} = \dfrac{\vec{F}_{net}}{m}$$ | ||
- | Notice that this means that the acceleration of system always points in the direction of the net force (because mass is always a positive quantity). | + | //Notice that this means that the acceleration of system always points in the direction of the net force (because mass is always a positive quantity).// |
- | It can also be defined (as above) in terms of the change in velocity over time. If this change is calculated over a time interval ($\Delta t$), then you obtain the //average// acceleration, | + | It can also be defined (as above) in terms of the change in velocity over time. If this change is calculated over a time interval ($\Delta t$), then you obtain the //average acceleration, |
$$\vec{a}_{avg} = \dfrac{\Delta \vec{v}}{\Delta t} = \dfrac{\vec{v}_f - \vec{v}_i}{\Delta t}$$ | $$\vec{a}_{avg} = \dfrac{\Delta \vec{v}}{\Delta t} = \dfrac{\vec{v}_f - \vec{v}_i}{\Delta t}$$ | ||
- | If we allow the time interval to shrink ([[: | + | If we allow the time interval to shrink ([[: |
+ | // | ||
$$\vec{a} = \lim_{\Delta t \rightarrow 0}\vec{a}_{avg} = \lim_{\Delta t \rightarrow 0}\dfrac{\Delta \vec{v}}{\Delta t} = \dfrac{d\vec{v}}{dt}$$ | $$\vec{a} = \lim_{\Delta t \rightarrow 0}\vec{a}_{avg} = \lim_{\Delta t \rightarrow 0}\dfrac{\Delta \vec{v}}{\Delta t} = \dfrac{d\vec{v}}{dt}$$ | ||
- | The units of acceleration are meters per second per second ($\dfrac{m}{s^2}$). | + | |
==== Why not just use change in momentum? ==== | ==== Why not just use change in momentum? ==== | ||
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If you have one way of describing motion (i.e., using the concept of a change in momentum), why should you learn about acceleration? | If you have one way of describing motion (i.e., using the concept of a change in momentum), why should you learn about acceleration? | ||
- | **Finish | + | Acceleration is a useful concept in mechanics, because it can help characterize the motion of systems (e.g., constant velocity motion has no acceleration). |
+ | |||
+ | While you can obtain |