Question

the velocity (v) of a body moving along positive x-direction varies with displacement (x) as v=krootx, where k is a constant . Determine how (v) varies with time (t) . Also , find the variation of x and t.

Answer 1

Dear Student,
$$\begin{gathered} heregiven \\ v=k\sqrt{x}-----1 \\ or{v}^2=k^{2}x \\ differentiatewithrespecttoxweget \\ \frac{2vdv}{dx}=k^2 \\ or \\ 2vdv=k^{2}dx\frac{dt}{dt}=k^{2}vdt \\ {\int }_0^{v}2dv={\int }_0^t{k}^{2}dt \\ v=k^{2}t/2 \\ fromequation1 \\ \frac{dx}{dt}=k\sqrt{x} \\ {\int }_0^x\frac{dx}{\sqrt{x}}={\int }_0^{t}kdt \\ -2\sqrt{x}=kt \\ x=\frac{k^2{t}^2}{4} \end{gathered}$$
Regard