Computer Programming/Physics/Position of an accelerating body function

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The position of an accelerating body can be described by a mathematical function $𝐬(t)$. The generalized function can be attained by using the Taylor series

$𝐬(t+t_0)=\sum \frac{t^n}{n!}{𝐬}^{(n)}(t_0)$,

where ${𝐬}^{(n)}$ is the $nth$ derivative of $𝐬$:

• ${𝐬}^{(0)}=\frac{d^0𝐬}{d{t}^0}=𝐬$
• ${𝐬}^{(1)}=\frac{d^1𝐬}{d{t}^1}=𝐯$
• ${𝐬}^{(2)}=\frac{d^2𝐬}{d{t}^2}=𝐚$
• etc.

The accuracy of this function depends on the number of terms used as $\frac{1}{n!}$ decreases rapidly. Additionally, the time $t$ can be synchronized such that $t_0=0$ (Maclaurin series).

Note that for a constant acceleration most of the terms become zero and we're left with

$s(t)=\frac{1}{0!}{𝐬}_0+\frac{t}{1!}{𝐬}_0^{(1)}+\frac{t^2}{2!}{𝐬}_0^{(2)}$

or

$𝐬(t)={𝐬}_0+{𝐯}_{0}t+\frac{1}{2}𝐚t^2$

C++

template<class Vector,class Number>
Vector PositionAcceleratingBody(Vector *s0,Number t,size_t Accuracy)
{
     Vector s(0);     //set to zero if int, float, etc. or invoke the
                      //     "set to zero" constructor for a class
     Number factor(1);//0!==1 and t^0==1
     for(size_t n(0);n<Accuracy;n++)
     {
          if(n)factor*=(t/n);//0!==1 and t^0==1
          s+=(factor*s0[n]); //s0 is the array of nth derivatives of s
                             //     at t=t0=0
     }
     return s;
}

Justification for Using the Taylor Series

The Taylor series can be derived by systematically selecting which of our variables is a constant and then extrapolating that to the infinite limit.

• Constant Position

$s(t)=s_0$

or

$s(t)=\frac{t^0}{0!}{s}^{(0)}(t_0=0)$

• Constant Velocity

$v(t)=v_0$

$s(t)=\int vdt=\int v_{0}dt$

$s(t)=v_{0}t+s_0$

or

$s(t)=\frac{t^0}{0!}{s}^{(0)}(t_0=0)+\frac{t^1}{1!}{s}^{(1)}(t_0=0)$

• Constant Acceleration

$a(t)=a_0$

$v(t)=\int adt=\int a_{0}dt$

$v(t)=a_{0}t+v_0$

$s(t)=\int vdt=\int (a_{0}t+v_0)dt$

$s(t)=\frac{1}{2}{a}_0{t}^2+v_{0}t+s_0$

or

$s(t)=\frac{t^0}{0!}{s}^{(0)}(t_0=0)+\frac{t^1}{1!}{s}^{(1)}(t_0=0)+\frac{t^2}{2!}{s}^{(2)}(t_0=0)$

• Constant Rate of Change of Acceleration

$a^{(1)}(t)=a_0^{(1)}$

$a(t)=\int a^{(1)}dt=\int a_0^{(1)}dt$

$a(t)=a_0^{(1)}t+a_0$

$v(t)=\int adt=\int (a_0^{(1)}t+a_0)dt$

$v(t)=\frac{1}{2}{a}_0^{(1)}{t}^2+a_{0}t+v_0$

$s(t)=\int vdt=\int (\frac{1}{2}{a}_0^{(1)}{t}^2+a_{0}t+v_0)dt$

$s(t)=\frac{1}{6}{a}_0^{(1)}{t}^3+\frac{1}{2}{a}_0{t}^2+v_{0}t+s_0$

or

$s(t)=\frac{t^0}{0!}{s}^{(0)}(t_0=0)+\frac{t^1}{1!}{s}^{(1)}(t_0=0)+\frac{t^2}{2!}{s}^{(2)}(t_0=0)+\frac{t^3}{3!}{s}^{(3)}(t_0=0)$

• etc.