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

Question

obwiertzy5w · Accepted Answer

The instantaneous velocity at any given point in a graph of position versus time

p (x, t)

on the function

x (t)

is the derivative of the function

x (t)

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} (t_{0}, x_{0})

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 = m x + 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 (t) = m t + 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 (t) = 2 t + 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

(x_{0}, t_{0})

, 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 (t) = t^{2}

) this would not be the case, and we would need to use differentiation techniques to find the derivatives of such functions.

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

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