Question

plz solve fast its urgent

Answer 1

LET'S

FOR CAR "A" => TIME =T1
____________VELOCITY = V1
____________ACCELERATION = A1
__________INITIAL VELOCITY = U1 = 0
_________________S1 = S

AND

FOR CAR "B" => TIME = T2
-___________VELOCITY = V2
-__________ACCELERATION = A2
-_______INITIAL VELOCITY = U2 = 0
-_________________S2 = S

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IT IS GIVEN THAT :-
$= > T _{1} + T = T _{2} \\ \\ AND \\ \\ = > V _{1} = V _{2} + V$
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NOW

BY EQUATION OF MOTION :-

$= > S _{1} = U _{1}T _{1} + \frac{1}{2} A _{1} {T_{1} }^{2} \\ \\ = > S = \frac{1}{2} A _{1} {T _{1} }^{2} \: \: \: \: ...[Eq _{1}]$
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$= > S _{2} = U _{2}T _{2} + \frac{1}{2} A _{2} {T _{2} }^{2} \: \: \\ = > S = \frac{1}{2} A _{2} {T _{2}}^{2} \: \: \: \: \: ...[Eq _{2}]$
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FORM Eq(1) and Eq(2)

$= > \frac{1}{2} A _{1} {tT _{1} }^{2} = \frac{1}{2} A _{2} {T _{2} }^{2} \\ \\ = > \frac{T _{2} }{T _{1} } = \sqrt{ \frac{A _{1} }{A _{2} } } \\ \\ = > \frac{T _{1} + T }{T _{1} } = \sqrt{ \frac{A _{1} }{A _{2} } } \\ \\ = > 1 + \frac{T}{T _{1} } = \sqrt{ \frac{A _{1} }{A _{2} } } \: \: \: ...[Eq _{3}]$
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$= > V _{1} = U _{1} + A _{1} T _{1} \\ \\ = > V _{1} = A _{1}T _{1} \: \: \: \: \: ...[Eq _{4}] \\ \\ AND \\ \\ = > V _{2} = U _{2} + A _{2}T _{2} \\ \\ = > V _{2} = A _{2}T _{2} \: \: \: \: \: ...[Eq _{5}]$

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$= > {V _{1} }^{2} = {U _{1} }^{2} + 2A _{1}S \\ \\ = > {V _{1} }^{2} = 2A _{1}S \: \: \: \: ...[Eq _{6}] \\ \\ AND \\ \\ = > {V _{2} }^{2} = {U _{2} }^{2} + 2A _{2}S \\ \\ = > {V _{2} }^{2} = 2A _{2}S \: \: \:... [Eq _{7}]$
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BY Eq(6) AND Eq(7)

$= > \frac{V _{1} }{V _{2}} = \sqrt{ \frac{A _{1} }{A _{2} } } \\ \\ = > \frac{V _{2} + V}{V _{2} } = \sqrt{ \frac{A _{1} }{A _{2} } } \\ \\ = > 1 + \frac{V}{V _{2} } = \sqrt{ \frac{A _{1} }{A _{2} } } \: \: \: \: ...[Eq _{8} ]$
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BY Eq(3) and Eq(8) , WE GET

$= > \frac{V _{1} }{V _{2} } = \frac{T}{T _{1} } \\ \\ = > \frac{V}{A _{2}T _{2} } = \frac{T}{ T_{1} } \\ \\ = > V = TA _{2} \frac{T _{2} }{ T_{1} } = TA _{2} \sqrt{ \frac{A _{1} }{A _{2} } } \\ \\ = > \frac{V}{T} = \sqrt{A _{1}A _{2} }$
________________[PROVED]

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_____☆$BE \: \: \: BRAINLY$☆_____