= Solution
Let the particle have <mass> $m$, and assume its <force> is derived from a time-independent <potential energy> $V(x)$ during the stated interval. Differentiation gives
$$
\dot x=nAt^{n-1},\qquad \ddot x=n(n-1)At^{n-2}.
$$
For $n\ne0$, choose a branch on which $t=(x/A)^{1/n}$ is real. <Conservation of energy> is the quickest integration of <Newton's second law>:
$$
\frac12m\dot x^2+V(x)=E,
\qquad
\boxed{V(x)=E-\frac{mn^2A^2}{2}\left(\frac{x}{A}\right)^{2-2/n}.}
$$
The additive constant $E$ is arbitrary. For instance, the displayed expression applies directly for $t>0$, $A\ne0$ and $x/A>0$, without raising a possibly negative $A$ to a fractional power. Its <derivative> gives $-V'(x)=mn(n-1)At^{n-2}$ along the trajectory, as required. The second term has units of <kinetic energy> because $A$ has units $\mathrm{m\,s^{-n}}$.
For $n=1$, the <velocity> is constant, so the <potential energy> is constant on the visited interval; the expression above reduces to such a constant. For $n=0$, or $A=0$, the particle is stationary: the trajectory only fixes $V'$ to vanish at its position and does not determine the <potential energy> elsewhere. These exceptional cases cannot determine a unique nonconstant function.
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