If the applied external force is zero, we 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

- A nonzero initial position and zero initial velocity results in for all (mass ``just sits there'')
- Similarly, any initial velocity 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).

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