My question involves simple math, but to be precise on what I'm asking, I need to write a lengthy description.
Let us define the following symbols:
$t$: time
$x(t)$: distance as a function of time $t$
$v(t)=\frac{dx(t)}{dt}$: velocity as a function of time $t$
$a(t)=\frac{dv(t)}{dt}$: acceleration as a function of time $t$
$v_f$: final velocity
$v_0$: initial velocity
$x_f$: final distance
$x_0$: initial distance
$t_f$: final time
$t_0$: initial time
$\Delta t$=$t_f$-$t_0$
$\Delta x$=$x_f$-$x_0$
For constant acceleration, the following well-known formulas hold and can derived by integrating $a(t)$ (they are special cases of Find Distance Function from Acceleration Function):
$v_f-v_0=a\Delta t$,
$\Delta x=v_0*\Delta t+\frac{1}{2}a\Delta t^2$
Combining them we get another well-known formula for constant acceleration:
$v_f^2=v_0^2+2*a*\Delta x$
My question is concerned with non-constant acceleration. In a scenario in which $a(t)$ is not a constant function, given $v_f$, $v_0$, and $\Delta x$, let us use the last equation to solve for an effective acceleration, $a_{eff}$:
$v_f^2=v_0^2+2*a_{eff}*\Delta x$
$a_{eff}$ is some sort of an average acceleration. I'd like to know exactly what kind of average $a_{eff}$ is.
Let us assume $x(t)$ is a one-to-one, but non-linear function of $t$. So we can define an inverse function $t(x)$, which gives the time as a function of distance.
Claim: $$a_{eff} = \frac{\int_{x_0}^{x_f} a(t(x)) dx}{ \Delta x}$$
In other words, $a_{eff}$ is the distance-averaged acceleration (which is not equal to the time-averaged acceleration because $x(t)$ is non-linear).
Is this claim true? My TA claimed it is, but was unwilling to supply a proof.
Best Answer
Notation such as $a(t)$ is fine and well for some purposes but it tends to make things more complicated when you want to average things over a distance.
Assuming the function from time $t$ to distance $x(t)$ is invertible, the object in question has a unique acceleration at each point along its path. You write $a(t(x))$ for the acceleration, but you can compose the two functions in that expression into a single function $a_x$, where $a_x(x) = a(t(x)).$
If you also assume that the accelerated object has mass $m$, then $ma_x(x)$ is the net force on the object at each point along its path. In that case, the work done on the object in traveling from $x_0$ to $x_f$ is simply $$\Delta E = \int_{x_0}^{x_f} ma_x(x)\, dx = m \int_{x_0}^{x_f} a_x(x)\, dx.$$ If we wanted to do the same amount of work on the object using a constant force $F_{\mathrm const}$ at each point along that segment (that is, using a constant acceleration $a_{\mathrm const} = \frac{F_{\mathrm const}}m$), the work done would be $$\Delta E = ma_{\mathrm const} (x_f - x_0) = ma_{\mathrm const} \Delta x.$$ Together, these equations imply that $$ma_{\mathrm const} \Delta x = m \int_{x_0}^{x_f} a_x(x)\, dx.$$ For $m \neq 0$ and $\Delta x \neq 0,$ and writing $a_{\mathrm eff} = a_{\mathrm const}$ for the constant acceleration, this is equivalent to your equation.
That is, as long as you can express the acceleration as a function of position along the path, the $a_{\mathrm eff}$ in $v_f^2 = v_0^2 + 2a_{\mathrm eff}\Delta x$ is acceleration averaged over distance.
More generally, it is not even necessary for $x(t)$ to be invertible, as long as it is still possible to write the acceleration as a function of distance $a_x(x)$ rather than time. The function $a_x(x)$ then describes a conservative force field, and the integral $\int a_x(x)\, dx$ measures the potential of this field at each point. The formula $$a_{\mathrm eff} = \frac{1}{\Delta x} \int_{x_0}^{x_f} a_x(x)\, dx$$ is still valid in that case, although it is not valid to write $a(t(x)).$ For example, you could initially set the object moving in the direction opposite to the direction of acceleration, in which case (if the acceleration continues to act in the same direction and continues to be strong enough, though not necessarily constant), the object will eventually come to a stop and then start traveling back to where it came from. In that case the function $x(t)$ is definitely not invertible, but $v_f^2 - v_0^2$ will still be determined by the locations (values of $x$) at which $v_0$ and $v_f$ are measured.
Edit:
We can derive the potential of a force field without invoking mass at all. The "force" field really is just a field of acceleration vectors; these create force when you introduce an object with mass, but the field potential is defined mathematically even if the masses of objects are never mentioned. Let $a$ be the acceleration at any given location (the acceleration vector at that location as defined by the "force" field), and let $v$ be the velocity of a particle acted upon by that acceleration. As usual, $v = \frac{dx}{dt}$ and $a = \frac{dv}{dt}$. Then $$a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt} = v \frac{dv}{dx},$$ and $$\int a\,dx = \int v \frac{dv}{dx} dx = \int v\, dv = \frac12 v^2.$$ Over a particular segment of the path from $x_0$ to $x_f$, the potential of the field is converted into velocity according to the following equation obtained by taking a definite integral of $a$ rather than an indefinite integral: $$\frac12 v_f^2 - \frac12 v_0^2 = \int_{x_0}^{x_f} a\, dx.$$ If we then find the average acceleration with respect to distance, $$a_{\mathrm avg} = \frac{1}{\Delta x} \int_{x_0}^{x_f} a\, dx,$$ where $\Delta x = x_f - x_0,$ then $$v_f^2 = v_0^2 + 2a_{\mathrm avg} \Delta x.$$