question_answer

A body of mass M is moved along a straight line by a machine delivering a constant power P. Find the expression for the distance moved by the body in terms of M, P and t.            

Answer:

                Power,  \[P=F\upsilon =ma.\upsilon =m\frac{d\upsilon }{dt}.\upsilon \]                 or            \[\upsilon d\upsilon =\frac{P}{m}dt\] Integrating both sides, we get \[\int{\upsilon dt}=\int{\frac{P}{m}dt}=\frac{P}{m}\int{dt}\]  [\[\because P,m\]are constant ] or            \[\frac{{{\upsilon }^{2}}}{2}=\frac{P}{m}.t\] or            \[{{\upsilon }^{2}}=\frac{2P}{m}.t\] or            \[\upsilon =\sqrt{\frac{2P}{m}.{{t}^{1/2}}}\] Thus      \[\upsilon \propto {{t}^{1/2}}\] Also       \[\upsilon =\frac{dx}{dt}\] or \[dx=\upsilon dt=\sqrt{\frac{2P}{m}}.{{t}^{1/2}}dt\] Integrating both sides , we get  \[\int_{{}}^{{}}{dx=\sqrt{\frac{2P}{m}}}\int_{{}}^{{}}{{{t}^{1/2}}dt}\] or            \[x=\sqrt{\frac{2P}{m}}.\frac{{{t}^{3/2}}}{3/2}=\frac{2}{3}\sqrt{\frac{2P}{m}}.{{t}^{3/2}}\] Thus \[x\propto {{t}^{3/2}}\]