The acceleration of a particle moving along x- axis is a= -100x+50. It

1.	The acceleration of a particle moving along x- axis is a= -100x+50. It is released from x=2. Here a and x are in SI units. The speed of the particle at origin will be......
Answer» Answer: $a = 50 - 100x$ $= > <klux>V</klux> \dfrac{dv}{dx} = 50 - 100x$ $= > v \: dv = (50 - 100x)dx$ Integrating on both sides : $= > \int v \: dv = \int \: (50 - 100x)dx$ APPLYING GIVEN limits ( i.e. at X = 2m , the Velocity was 0 , and at x = 0 m , Velocity be v) $= > \int_{0}^{v} v \: dv = \int_{2}^{0} \: (50 - 100x)dx$ $= > \bigg \{ \dfrac{ {v}^{2} }{2} \bigg \}_{0}^{v} = \bigg \{50x - 50 {x}^{2} \bigg \}_{2}^{0}$ $= > \dfrac{ {v}^{2} }{2} = \{0 - (50 \times 2) \} - \{0 - (50 \times {2}^{2}) \}$ $= > \dfrac{ {v}^{2} }{2} = 200 - 100$ $= > \dfrac{ {v}^{2} }{2} = 100$ $= > {v}^{2} = 200$ $= > v = \sqrt{200}$ $= > v = 10 \sqrt{2} \: m {s}^{ - 1}$ So final answer : $\boxed{ \boxed{\sf{ \red{ \large{Velocity_{(at \: x = 0 \: m)} = 10 \sqrt{2} \: m {s}^{ - 1} }}}}}$