 Lexi Holmes

2023-03-24

How to find instantaneous velocity from a position vs. time graph? obwiertzy5w

The instantaneous velocity at any given point in a graph of position versus time $p\left(x,t\right)$ on the function $x\left(t\right)$ is the derivative of the function $x\left(t\right)$ with respect to time at that point.
The derivative of a function at any given point is simply the instantaneous rate of change of the function at that point. In the case of a graph of position (or distance) vs. time, that means that the derivative at a given point ${p}_{0}\left({t}_{0},{x}_{0}\right)$ is the instantaneous rate of change in position (accounting for ""positive"" and ""negative"" direction) with respect to time.
As an example, consider a linear distance function (that is, one which can be represented with a line as opposed to a curve). If this were a function of $x$ and $y$, with $y$ as the dependent variable, then our function in slope-intercept form would take the form $y=mx+b$, where $m$ is the slope and $b$ is the value of $y$ at $x=0$. In this case, $t$ is our independent variable and $x$ is our dependent, so our linear function would take the form $x\left(t\right)=mt+b$.
From algebra, we know that the slope of a line measures the number of units of change in the dependent variable for every single unit of change in the independent variable. Thus, in the line $x\left(t\right)=2t+5$, for every one unit by which $t$ increases, $x$ increases by 2 units. If we were to, for example, assign units of seconds to $t$ and feet to $x$, then every second that passed (that is, every increase of one second in $t$), position (or distance) would increase by two feet (that is $x$ would increase by two feet)
Since our change in distance per unit of change in time will remain the same no matter our starting point $\left({x}_{0},{t}_{0}\right)$, in this case we can be assured that our instantaneous velocity is the same throughout. Specifically, it is equal to $m=2$. Differentiating the function with respect to $t$ yields the same answer. Note that this is only identical to our average velocity throughout the function by design: for a non-linear function (such as $x\left(t\right)={t}^{2}$) this would not be the case, and we would need to use differentiation techniques to find the derivatives of such functions.

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