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A particle of mass m moves in a straight line with its velocity varying with distance travelled according to the equation $v=a\sqrt {x}$ , where a is a constant. What is the total work done by all the forces during a displacement from x = 0 to x = d?

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Answer: $W = \large\frac{1}{2}$$ma^2d$
Work done due to a constant force $W = Fd$.
Constant Force = mass $\times$ acceleration.
Acceleration is given by $ \large\frac{dv}{dt}$$ = \large\frac{dv}{dx}$$ \large\frac{dx}{dt}$
$\Rightarrow = \large\frac{dv}{dt}$$ = \large\frac{a}{2\sqrt {x}}$$a \sqrt{x}$$ = \large\frac{1}{2}$$a^2$
$\Rightarrow W = \large\frac{1}{2}$$ma^2d$
answered Aug 25, 2014 by balaji.thirumalai

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