I'm trying to find a function $f$ that simulates the motion of a particle that conforms to the following constraints.
The particle begins at position 0 and moves to 1.
The velocity is 0 at the start and end of the motion.
The particle accelerates at a constant rate until an inflection point and then decelerates at a constant rate.
Because $f(0)=0$, $f'(0)=0$, and $f''(0)=0$, I start with $x^3$ and then find a function $g$ that satisfies the other conditions. First take the derivatives.
$$f(x)=x^3g(x)$$
$$f'(x)=3x^2g(x)+x^3g'(x)$$
$$f''(x)=6xg(x)+6x^2g'(x)+x^3g''(x)$$
Then solve for the values of $g$ that correspond to the points I want.
$$f'(1)=0 \Rightarrow 3g(1)+g'(1)=0 \Rightarrow g'(1)=-3$$
$$f''(1)=0 \Rightarrow 6g(1)+6g'(1)+g''(1) \Rightarrow g''(1)=12$$
Then, choosing $g(x)=ax^2+bx+c$, solve for the constants.
$$g''(1)=12 \Rightarrow 2a=12 \Rightarrow a=6$$
$$g'(1)=-3 \Rightarrow 2a+b=-3 \Rightarrow b=-15$$
$$g(1)=1 \Rightarrow a+b+c=1 \Rightarrow c=10$$
This gives $f(x)=x^3(6x^2-15x+10)=6x^5-15x^4+10x^3$. This function works when the inflection point is at $x=0.5$.
What I want is a way to find a function for any inflection point in $(0,1)$. I'm not sure how to generalize this process or expand on it to accomplish that.
The argument of the function should be the time t. It can be assumed that the particle reaches the point 1 at time T which is a parameter. Now we have $x(0)=0 ;\; x(T)=1;\; \dot x(0)=0;\; \dot x(T)=0\;.$ Furthermore, because of symmetry, we have $\ddot x(t)=a$ on $0 \le t \lt T/2$ and $\ddot x(t)=-a$ on $T/2 \le t \le T.$ The intervals $0 \le t \lt T/2$ and $T/2 \le t \le T$ can now be treated separately. Integration in the first interval gives: $\dot x=at+v_0\;$ where $v_0=0$ because of $\dot x(0)=0$. Another integration gives $x(t)=\frac{a}{2}t^2+x_0$ where $x_0=0$ because of $x(0)=0$. The second interval can be treated analogously to find the solution $x(t)=-\frac{a}{2}t^2+aTt+1-\frac{a}{2}T^2$. (It is a little more laborious to determine the constants $x_1$ and $v_1$ here.) By derivation and inserting the given values at time 0 and T it can be checked that the required conditions are fullfilled. If the inflection point is not at $T/2$, replace $T/2$ by a parameter value, say $T_i$, and use different accelerations, e.g. $a_1;\;-a_2$, in both intervals.