A particle is projected with velocity V_0 along X - axis . The deceleration on the particle is proportional to the square of the distance from the origin i.e., \(a = \alpha x^{2} .\) The distance at which the particle stops is
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\(a = \frac{dv}{dt} = \frac{dv}{dx} \frac{dx}{dt}\) \(V \frac{dv}{dx} = - \alpha x^{2}\) (given)
⇒ ⇒ \(\int_{v_0}^{0} v \, dv = -\alpha \int_{0}^{s} x^{2} \, dx\) ⇒ ⇒ \(\left[\frac{\mathbf{v}^2}{2}\right]_0^{v_0} = -\alpha \left[\frac{\mathbf{x}^3}{3}\right]_0^s\)
⇒ ⇒ \(\frac{v_0^2}{2} = \frac{\alpha S^3}{3}\) ⇒ ⇒ \(S = \left( \frac{3v_0^2}{2\alpha} \right)^{\frac{1}{3}}\)
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