Question

CBSE class 11 A particle located at x= 0 at time t= 0 starts moving along the positive x direction with a velocity v that varies as v = α√x . The displacement of the particle ovaries with time as 1)$t^{\frac{1}{2} }$ 2)t³ 3)t² 4)t I need the step by step explanation. Please no plagiarism from net or absurd answers like " i dont know". please help...

Answer 1

at x= 0 and t = 0

$v = a \sqrt{x}$

$\frac{dx}{dt} = p \sqrt{x}$

$\lim_{0 \to \ x} \frac{dx}{ {x}^{ \frac{1}{2} } } = a\lim_{0 \to \ t}dt$

$\frac{ {x}^{ \frac{1}{2} } { \frac{1}{2} } = at$

$2 {x}^{ \frac{1}{2} } = at$

${x}^{ \frac{1}{2} } = \frac{at}{2}$

$x = { \frac{at}{2} }^{2}$

$x = \frac{ {a}^{2} {t}^{2} }{4}$

$x \alpha {t}^{2}$