To find the dimensions of energy when force $[F],$ acceleration $[A],$ and time $[T]$ are the fundamental physical quantities, we start by recalling the formula for energy.$\\$Energy (specifically kinetic energy) is given by:$\\ E = \frac{1}{2} m v^2 \\$We need to express mass $(m)$ and velocity $(v)$ in terms of $[F], [A],$ and $[T]. \\$From Newton's second law, we have:$\\ F = m \cdot A \\$Thus, mass $m$ can be expressed as:$\\ m = \frac{F}{A} \\$Velocity $v$ is defined as distance over time, but we can also express it in terms of acceleration and time:$\\ v = A \cdot T \\$Substitute these expressions into the energy formula:$\\ E = \frac{1}{2} \left(\frac{F}{A}\right) (A \cdot T)^2 \\$Simplify the expression:$\\ E = \frac{1}{2} \left(\frac{F}{A}\right) (A^2 \cdot T^2) \\ E = \frac{1}{2} F \cdot A \cdot T^2 \\$Since the constant $\frac{1}{2}$ does not affect dimensions, the dimensional formula for energy is:$\\ [F] [A] [T^2] \\$Therefore, the correct option is Option 2.