The image above is from Crash Course and I saw the equation the velocity is Delta.
What does the line above acceleration represent?
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$\begingroup$ Average value.... Although there are different conventions, so that is just my interpretation $\endgroup$– user163104Commented Aug 7, 2017 at 6:04
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$\begingroup$ The velocity is not Delta. The velocity is $v$. The Delta symbol $\Delta$ means difference. So $\Delta v$ is the difference in velocity. You would calculate it like: $$\Delta v=v_{\text{after}}-v_{\text{before}}$$Likewise, $\Delta t$ is the difference in time. $\endgroup$– SteevenCommented Aug 7, 2017 at 6:54
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$\begingroup$ Means what @Steeven? $\endgroup$– kartladCommented Aug 7, 2017 at 6:54
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2 Answers
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Usually, a bar over a symbol means its average. Also, one can use the symbols $\left< - \right>$ can be used to denote the same thing.
Usually, this average is a time average, that is
$$\left< s \right> = \frac{1}{t_1-t_0}\int\limits_{t_0}^{t_1} s(t)\textrm{d}t$$
where either $t_0 \rightarrow -\infty$ and $t_1 \rightarrow +\infty$, or $t_0$ is the start of the experiment and $t_1$ its end.
In your case, assuming the experiment starts at $t=0$ and lasts $\Delta t$, it gives
$$\left< a \right> = \frac{1}{\Delta t}\int\limits_0^{\Delta t} a(t)\textrm{d}t = \frac{1}{\Delta t} \int\limits_0^{\Delta t} \frac{\textrm{d}v}{\textrm{d}t}\textrm{d}t = \frac{v(\Delta t) - v(0)}{\Delta t} = \frac{\Delta v}{\Delta t}$$
Notice the use of the symbol $\Delta$ to denote a variation of a function. Here for example $\Delta v$ means the variation of the function $v$ of variable $t$. You could also encounter the symbol $\textrm{d}v$: both $\Delta$ and $\textrm{d}$ mean the same thing (ie. a variation of something), however $\textrm{d}$ means that it is an infinitesimal variation. From a mathematical point of view, this is complete nonsense: what is a difference between a big, a normal, and a small variation ? And what do big, small mean ? In fact, the difference is that $\textrm{d}$ implies implicitly a limit, while $\Delta$ is just a normal quantity. Then whenever you see a $\textrm{d}s$ instead of a $\Delta s$, you should interpret it as when the variation of $s$ tend toward zero, then ... You can now understand the notation $\frac{\textrm{d}}{\textrm{d}t}$ to denote the temporal derivative. Let's consider a function $f(t)$, then
$$f'(t) = \lim_{\textrm{d}t\to 0} \frac{f(t+\textrm{d}t) - f(t)}{\textrm{d}t} = \frac{\textrm{d}f}{\textrm{d}t}$$
Similarly, when you are referring to a small quantity of something, and not a variation of something, you should use the symbol $\delta$ (which implies a limit) and no symbol at all when the quantity isn't small. For example, if you consider the heat transferred during a whole experience, you could call it $Q$, while during a infinitesimal transformation it should be $\delta Q$
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$\begingroup$ +1 Could you incorporate @Steeven comment about $\Delta v$ in your answer, as comments get wiped and, I am pushing here, also refer to the phrase expectation value, if you have time? Just a thought for future OPs, with similar questions $\endgroup$– user163104Commented Aug 7, 2017 at 8:32
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The bar over the $'a'$ means average acceleration. It's not a typo in the video. A bar above $'v'$ would denote average velocity.
A bar above any quantity indicates that it is the average value of that quantity.
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$\begingroup$ So, the bar above any character represents change? $\endgroup$– kartladCommented Aug 7, 2017 at 6:08
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$\begingroup$ It represents average not change. $\endgroup$– MitchellCommented Aug 7, 2017 at 6:14
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$\begingroup$ You say delta, but delta V is the change in V $\endgroup$– user163104Commented Aug 7, 2017 at 6:17