The distance x covered by a particle in one dimensional motion varies with time t as ${\mathrm{x}}^2={\mathrm{at}}^2+2\mathrm{bt}+\mathrm{c}.$. If the acceleration of the particle depends on x as x^-n, where n is an integer, the value of n is $\_\_\_\_\_\_$

Distance x varies with time t as,

x² = at² + 2bt +c ...(1)

Differentiating both sides with respect to t,

⇒ 2xdx/dt = 2at + 2b

⇒ xdx/dt = at + b ...(2)

Again differentiating,

⇒ xd²x/dt² + (dx/dt)² = a

⇒ d²x/dt² = [a − (dx/dt)²]/x

Using equation (2),

⇒ d²x/dt² = [a −( (at + b)/x)²]/x

⇒ a = d²x/dt² = [ax² − (at+b)²]/x³

Using equation (1),

⇒ a = [ac − b²]/x³

⇒ a ∝ x⁻³

Hence, n=3