I'm trying to generate a transmission function from the distance between to moving particles (or coordinate systems). Below is a plot, $D$ being the distance between the two particles:
The two particles are moving according to:
$$ d_1(t)=At^3$$ $$ d_2(t)=-Mt+b$$
I want to create a transmission function, say $F_T$ that is a function of the distance between the two particles, $D$, and has the following boundary conditions:
$$ F_T(t=0) = \frac{c}{b}$$ $$ F_T(t=t_{collision}) = 1 $$
where $t_{collision}$ is the time in which the particles collide and $D=0$. The function would only need to be valid during positive times before the collision and we need not worry about times (or dynamics) after.
I thought of trying this a few ways. First, was to find $D=-Mt+b-At^3$ and to guess at forms of $F_T$ that would satisfy its boundary conditions. So, for example:
$$F_T(t)= \frac{c}{b^2} \left( D(t) \right)$$
which satisfies the $t=0$ boundary but not the $t_{collision}$ boundary. But, no matter what I've guessed I haven't been able to satisfy both boundary conditions!
So, essentially I want a transmission function, $F_T$, that will non-linear behavior of the distance, $D$, between the two coordinate systems and I thought there might be a more formal calculation rather than me guessing at forms. Can anyone be of any help!?
First start with a function of distance: $T(D)$ such that $T(0)=1$ and $T(b)=\frac cb$. You can have many choices, such as a linear function or an exponential. Let's choose an exponential. Since $T(0)=1$ we can write this as $$T(D)=e^{\lambda D}$$ Then knowing $T(b)=\frac cb$ we can find $\lambda$: $$e^{\lambda b}=\frac cb\\\lambda b=\ln\frac cb\\\lambda=\frac 1b\ln\frac cb$$ So:$$T(D)=e^{\frac Db\ln\frac cb}$$ Now what's left is to transform to $F_T(t)$, using $$F_T(t)=T(D(t))$$