Equations of motion (2nd order ODEs)

We know that the motion of an object is determined by Newton's equations. In the one-dimensional case, we can define the instantaneous position $y(t)$, velocity $v(t)$ and acceleration $a(t)$ of an object using the language of differential calculus:

$$v(t)=\frac{dy}{dt}, a(t)=\frac{dv}{dt}.$$ (19)

The motion of the particle is defined by:

$$\frac{d^2y}{dt^2}=\frac{F(t)}{m}$$

This is a second order differential equation that can written as two first order differential equations:

$$\frac{dv}{dt}=\frac{F(t)}{m};$$ (20)

$$\frac{dy}{dt}=v(t).$$ (21)

To solve it we can apply any of the methods described in the previous sections. If we pick Euler's, we obtain:

$$v_{n+1}=v_n+\frac{F(t)}{m}\Delta t = v_n+a_n\Delta t,$$ (22)

$$y_{n+1}=y_n+v_n\Delta t,$$ (23)

where $a_n=F(t)/m$.