The speed v reached by a car of mass m in travelling a distance x, driven with constant power P, is given by :
Answer & explanation
Correct answer: option 3
\(P = Fv = m\frac{dv}{dt} v\)
\(\Rightarrow v\frac{dv}{dt} = \frac{P}{m}\)
\(\Rightarrow v\frac{dv}{dx}{dx}{dt} = \frac{P}{m}\)
\(\Rightarrow v^2 \frac{dv}{dx} = \frac{P}{m}\)
\(\Rightarrow v^2 dv = \frac{P}{m} dx\)
\(\Rightarrow \int_0^v v^2 dv = \frac{P}{m} \int_0^x dx\)
\(\Rightarrow \frac{v^3}{3} = \frac{Px}{m}\)
\(\Rightarrow v = (\frac{3xP}{m})^{1/3}\)