A) 1.5 J
B) 50 J
C) 10 J
D) 100 J
Correct Answer: B
Solution :
[b]: Given : \[v=a{{x}^{3/2}}\]where,\[a=5{{m}^{-1/2}}{{s}^{-1}}\] Acceleration\[=\frac{dv}{dt}=\frac{dv}{dx}.\frac{dx}{dt}=v\frac{dv}{dx}\left( \because v=\frac{dx}{dt} \right)\] As\[{{v}^{2}}={{a}^{2}}{{x}^{3}}\] Differentiating both sides with respect to x, we get \[2v\frac{dv}{dx}=3{{a}^{2}}{{x}^{2}}\]or, Acceleration\[=\frac{3}{2}{{a}^{2}}{{x}^{2}}\] Force, F = Mass\[\times \]Acceleration \[=\frac{3}{2}m{{a}^{2}}{{x}^{2}}\] Work done, \[W=\int_{{}}^{{}}{Fdx}=\int\limits_{0}^{2}{\frac{3}{2}}m{{a}^{2}}{{x}^{2}}dx\] \[W=\frac{3}{2}m{{a}^{2}}\left[ \frac{{{x}^{3}}}{3} \right]_{0}^{2}=\frac{3}{2}\times 0.5\times {{5}^{2}}\times \frac{8}{3}=50J\]
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