Force-Driven Mass Analysis, Continued

If the applied external force $f(t)$ is zero, we obtain

$$X(s) = \frac{x(0)}{s} + \frac{{\dot x}(0)}{s^2} = \frac{x(0)}{s} + \frac{v(0)}{s^2}.$$

Since $1/s$ is the Laplace transform of the Heaviside unit-step function

$$u(t)\mathrel{\stackrel{\mathrm{\Delta}}{=}}\left\{\begin{array}{ll} 0, & t<0 \\ [5pt] 1, & t\ge 0 \\ \end{array} \right.,$$

we find that the position of the mass $x(t)$ is given for all time by

$$x(t) = x(0)\,u(t) + v(0)\,t\,u(t).$$

• A nonzero initial position $x(0)=x_0$ and zero initial velocity $v(0)=0$ results in $x(t)=x_0$ for all $t\ge 0$ (mass ``just sits there'')
• Similarly, any initial velocity $v(0)$ is integrated with respect to time (mass moves forever at initial velocity)

In summary, we used the Laplace transform to solve for the motion of a simple physical system (an ideal mass) in response to initial conditions (no external driving forces).