Question and Answers Forum

All Questions      Topic List

Mechanics Questions

Previous in All Question      Next in All Question      

Previous in Mechanics      Next in Mechanics      

Question Number 37579 by ajfour last updated on 15/Jun/18

$$a=\frac{-1}{2x^2}withx=1andv=0att=0$$$$findtimethatparticletakesto$$$$reachx=0.25m.$$

Answered by ajfour last updated on 15/Jun/18

$$sincea<0and{v}_0=0$$$$thismeansnegativevelocity$$$$startsdevelopingastincreases$$$$fromzero.$$$$\mathit{a}=\frac{\mathit{v}\mathit{d}\mathit{v}}{\mathit{d}\mathit{x}}=-\frac{1}{2{\mathit{x}}^2}$$$$\Rightarrow \frac{v^2}{2}=\frac{1}{2}(\frac{1}{x}-1)$$$$\Rightarrow v=-\sqrt{\frac{1}{x}-1}$$$${\int }_1^x\frac{dx}{\sqrt{\frac{1}{x}-1}}=-t$$$$letx=\cos {}^2\theta$$$$dx=-\sin 2\theta d\theta$$$$-t=-{\int }_0^{{\cos }^{-1}\sqrt{x}}\frac{2\sin \theta \cos \theta d\theta }{\tan \theta }$$$$t={\int }_0^{{\cos }^{-1}\sqrt{x}}(1+\cos 2\theta )d\theta$$$$=(\theta +\frac{\sin 2\theta }{2}+c){\mid }_0^{{\cos }^{-1}\sqrt{x}}$$$$t={\cos }^{-1}\sqrt{x}+\left(\sqrt{x}\right)\left(\sqrt{1-x}\right)$$$$Andtoreachx=0.25$$$$\mathit{t}=\frac{\mathit{\pi }}{3}+\frac{\sqrt{3}}{4}.$$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com