# 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?

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$