To find the minimum distance travelled by the particle, the function $s(t) = t^3 - t^2 - t + 5$ must be analyzed for its critical points by taking the derivative and setting it to zero. Solving the resulting equation reveals the values of $t$ that correspond to local minima or maxima. Evaluating the original function $s(t)$ at these critical points leads to determining the minimum distance from the fixed point.