Numerical solutions: step-by-stepAn object moving in one-dimension can be described in terms of its position $x$ and and its velocity $v_x$. If the force on the object is known, then the motion can be described by two first order differential equations, $\large \frac{dx}{dt}=v_x$ and $\large \frac{dv_x}{dt}=a_x=F_x(x,v_x,t)/m.$ Here $F$ is the force, $m$ is the mass, and $t$ is the time. These equations can be solved numerically. If the initial conditions for $x(t_0)$ and $v_x(t_0)$ are known, a good estimate for $x$ and $v_x$ a short time $\Delta t$ later is, $\large x(t_0+\Delta t) \approx \frac{dv_x}{dt}|_{t_0}\Delta t$ and $\large v_x(t_0+\Delta t) \approx F(x(t_0),v_x(t_0),t_0)\Delta t/m.$ Once an estimate for the position and the velocity of the object at time $t_0 + \Delta t$ is calculated, they can be used to estimate the position and the velocity at time $t_0 + 2\Delta t$. The solution is produced step-by-step as the time is increased by a small increment $\Delta t$. The form below can be used to numerically integrate these equations for a total number of $N_{steps}$ steps using a step size of $\Delta t$. The acceleration $a_x$ can be given as a function of $x$, $v_x$, and $t$. = , = |