Work Energy And Power Question 260
Question: A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time ‘f’ is proportional to
Options:
A) $ {t^{3/4}} $
B) $ {t^{3/2}} $
C) $ {t^{1/4}} $
D) $ {t^{1/2}} $
Show Answer
Answer:
Correct Answer: B
Solution:
[b] We know that $ F\times v = Power $
$ \therefore F\times v=c $ where c = constant $ a\therefore m\frac{dv}{dt}\times v=c~~~( \therefore F=ma=\frac{mdv}{dt} ) $
$ \therefore m\int\limits_0^{v}{vdv=c\int\limits_0^{t}{dt}\therefore \frac{1}{2}mv^{2}=ct} $
$ \therefore v=\sqrt{\frac{2c}{m}}\times {t^{{}^{1}/ _2}} $
$ \therefore \frac{dx}{dt}=\sqrt{\frac{2c}{m}}\times {t^{{}^{1}/ _2}}wherev=\frac{dx}{dt} $
$ \therefore \int\limits_0^{x}{dx=\sqrt{\frac{2c}{m}}\times \int\limits_0^{t}{{t^{{}^{1}/ _2}}}}dt $
$ x=\sqrt{\frac{2c}{m}}\times \frac{2{t^{{}^{3}/ _2}}}{3}\Rightarrow x\propto {t^{{}^{3}/ _2}} $
