A) \[{{t}^{\frac{1}{2}}}\]
B) \[{{t}^{\frac{3}{4}}}\]
C) \[{{t}^{\frac{3}{2}}}\]
D) \[{{t}^{2}}\]
Correct Answer: C
Solution :
We are given that a box is moved along a straight line by a machine under constant power. So, we have power P = Fv = mav \[P=m\frac{dv}{dt}v\] or \[\int_{{}}^{{}}{vdv}=\int_{{}}^{{}}{\frac{P}{m}dt}\] \[\frac{{{v}^{2}}}{2}=\frac{Pt}{m}\] or \[v=\sqrt{\frac{2P}{m}t}\] or \[\frac{dx}{dt}=\sqrt{\frac{2P}{m}t}\] \[\int_{{}}^{{}}{dx=\sqrt{\frac{2P}{m}}\int_{{}}^{{}}{{{t}^{1/2}}dt}}\] \[x=\sqrt{\frac{2P}{m}}\frac{{{t}^{3/2}}}{3/2}\] \[\therefore \] \[x\propto {{t}^{3/2}}\]
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