Determining acceleration in order to meet a space-time-velocity reservation

Question

An Example

A train on a single line track is told to be at position $x_{res}$, at time $t_{res}$, and to be moving at exactly $v_{res}$ when it hits that point. Where $_{res}$ denotes the reservation.

The train driver knows the current time ($t_{cur}$), it's current speed ($v_{cur}$) and it's current position ($x_{cur}$). Where $_{cur}$ denotes the current value.

Provide an equation that gives the acceleration the train driver must apply at $t_{cur}$, in order to be on track to meet the reservation.

The train could need to accelerate to faster than $v_{res}$ then slow down. It could instead have to constantly accelerate, or even maintain the exact same speed. It all depends on the situation.

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I'm looking to determine acceleration as a function of time, in order to meet a space-time-velocity reservation. - That is: be at location $x$, at time $t$, with final velocity $v$.

The problem domain can be considered 1 dimensional.

Available Values

• Current and final location ($x_1$ & $x_2$), thus $\Delta x$.
• Current and final time ($t_1$ & $t_2$), thus $\Delta t$
• Current and final velocity ($v_1$ & $v_2$), thus $\Delta v$

I am aware there are likely infinite acceleration curves that would technically solve this problem, an ideal solution would result in a curve with the least extreme accelerations.

The final answer should be an equation giving acceleration using the available values above. I would appreciate it if you could explain how you found your answer, and your patience with the limited knowledge of a laymen.

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What I've tried

I'm familiar with equations of motion under constant acceleration, however varied acceleration like this is still a little over my head. I have posted this question on Reddit, and received an answer, however that too is over my head, and I am unable to work through it, though I have attempted to. You can see this post and my attempts at working through it here

I am not a Physicist or a Mathematician, so please excuse any mistakes or misconceptions on my part. I would be truly grateful for any help.

Answer 1

You can't satisfy all four conditions

$$x(t_1)=x_1\tag 1$$ $$\dot{x}(t_1)=v_1\tag 2$$ $$x(t_2)=x_2\tag 3$$ $$\dot{x}(t_2)=v_2\tag 4$$

with a constant acceleration. But you can with a linearly changing acceleration of the form

$$\ddot{x}(t)=A+B(t-t_1).\tag 5$$

Integrating (5) and using condition (2) to determine the constant of integration, the velocity is

$$\dot{x}(t)=v_1+A(t-t_1)+\frac12B(t-t_1)^2.\tag 6.$$

Integrating (6) and using condition (1) to determine the constant of integration, the position is

$$x(t)=x_1+v_1(t-t_1)+\frac12A(t-t_1)^2+\frac16B(t-t_1)^3.\tag 7$$

Imposing condition (3) on (7) and condition (4) on (6) leads to two simultaneous equations to solve for $A$ and $B$. The solution is

$$A=2\frac{3(x_2-x_1)-(v_2+2v_1)(t_2-t_1)}{(t_2-t_1)^2}\tag 8$$

and

$$B=-6\frac{2(x_2-x_1)-(v_2+v_1)(t_2-t_1)}{(t_2-t_1)^3}.\tag 9$$