Figure D.1 depicts a free mass driven by an external force along an ideal frictionless surface in one dimension. Figure D.2 shows the electrical equivalent circuit for this scenario in which the external force is represented by a voltage source emitting volts, and the mass is modeled by an inductor having the value Henrys.
From Newton's second law of motion `` '', we have
Taking the unilateral Laplace transform and applying the differentiation theorem twice yields
If the applied external force is zero, then, by linearity of the Laplace transform, so is , and we readily obtain
Since is the Laplace transform of the Heaviside unit-step function
we find that the position of the mass is given for all time by
Thus, for example, a nonzero initial position and zero initial velocity results in for all ; that is, the mass ``just sits there''.D.3 Similarly, any initial velocity is integrated with respect to time, meaning that the mass moves forever at the initial velocity.
To summarize, this simple example illustrated use 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). The system was described by a differential equation which was converted to an algebraic equation by the Laplace transform.