Question

given that time t =ax^2 +bx, where a and b are constants. show that the retardation is 2av^3 where v is the instantaneous velocity.

Answer 1

\( t =ax^2 +bx\)
\(\frac{dt}{dx} = 2ax + b\) [or \(1 = 2ax \frac{dx}{dt} + b \frac{dx}{dt} = \frac{dx}{dt} (2ax+b)\) etc]
\(\frac{dx}{dt} = \frac{1}{2ax + b} [= v]\)
\(\frac{d^2x}{dt^2} = \frac{-2a}{(2ax + b)^2} [= -2av^2]\)

not \(v^3\)