A

Step 1: Given the displacement-time relation \( t = (x^2 - 1)^{\frac{1}{2}} \).
Step 2: Differentiate with respect to time to find velocity: \( \frac{dx}{dt} = \frac{d}{dt}[(x^2 - 1)^{\frac{1}{2}}] = \frac{1}{2}(x^2 - 1)^{-\frac{1}{2}} \cdot 2x \cdot \frac{dx}{dt} \).
Step 3: This simplifies to \( 1 = \frac{x \frac{dx}{dt}}{(x^2 - 1)^{\frac{1}{2}}} \).
Step 4: Rearranging gives: \( \frac{dx}{dt} = \frac{(x^2 - 1)^{\frac{1}{2}}}{x} \).
Step 5: Differentiate the velocity with respect to time to find acceleration: \( a = \frac{d}{dt}\left(\frac{(x^2 - 1)^{\frac{1}{2}}}{x}\right) = \frac{d}{dt}((x^2 - 1)^{\frac{1}{2}}) \cdot \frac{1}{x} - \frac{(x^2 - 1)^{\frac{1}{2}}}{x^2}\frac{dx}{dt} = \frac{(x) (x^2 - 1)^{-\frac{1}{2}} \cdot x\frac{dx}{dt}}{x} - \frac{\frac{(x^2 - 1)^{\frac{1}{2}}}{x^2} (x)}{x^{2}} \).
Step 6: The final expression gives acceleration in terms of displacement x.
Therefore, the acceleration in terms of x is captured in alternative forms which can be simplified further, resulting in the relation with respect to x.