a-1-2x-2-with-x-1-and-v-0-at-t-0-find-time-that-particle-takes-to-reach-x-0-25m-

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

a = \frac{- 1}{2 x^{2}} w i t h x = 1 a n d v = 0 a t t = 0

f i n d t i m e t h a t p a r t i c l e t a k e s t o

r e a c h x = 0.25 m .

Answered by ajfour last updated on 15/Jun/18

s i n c e a < 0 a n d v_{0} = 0

t h i s m e a n s n e g a t i v e v e l o c i t y

s t a r t s d e v e l o p i n g a s t i n c r e a s e s

f r o m z e r o .

a = \frac{v d v}{d x} = - \frac{1}{2 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{d x}{\sqrt{\frac{1}{x} - 1}} = - t

l e t x = \cos^{2} θ

d x = - \sin 2 θ d θ

- t = - \int_{0}^{\cos^{- 1} \sqrt{x}} \frac{2 \sin θ \cos θ d θ}{\tan θ}

t = \int_{0}^{\cos^{- 1} \sqrt{x}} (1 + \cos 2 θ) d θ

= (θ + \frac{\sin 2 θ}{2} + c) ∣_{0}^{\cos^{- 1} \sqrt{x}}

t = \cos^{- 1} \sqrt{x} + (\sqrt{x}) (\sqrt{1 - x})

A n d t o r e a c h x = 0.25

t = \frac{π}{3} + \frac{\sqrt{3}}{4} .

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