Work Energy Power

JEE Main level, single correct, medium

119 of 183

Question 119 of 183

A body of mass mm starts moving from rest along x−x-axis so that its velocity varies as v=asv=a\sqrt{s} where aa is a constant and ss is the distance covered by the body. The total work done by all the forces acting on the body in the first tt second after the start of the motion is

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Answer: (D) 18ma4t2\dfrac{1}{8}m{a}^{4}{t}^{2}

v=as=dsdtv=a\sqrt{s}=\dfrac{ds}{dt}

⇒dss=a.dt\Rightarrow \dfrac{\text{d}\text{s}}{\sqrt{\text{s}}}=a.dt

⇒2s=at\Rightarrow 2\sqrt{s}=at

⇒s=a2t24,v=dsdt=a2t2\Rightarrow s=\dfrac{{a}^{2}{t}^{2}}{4},v=\dfrac{ds}{dt}=\dfrac{{a}^{2}t}{2}

⇒Wall=∆K=Kf−Ki\Rightarrow {W}_{all}=∆K={K}_{f}-{K}_{i}

⇒Wall=12m[v22−v12]\Rightarrow {W}_{all}=\dfrac{1}{2}m\left[{v}_{2}^{2}-{v}_{1}^{2}\right]

⇒v1=0,v2=a2t2\Rightarrow {v}_{1}=0,{v}_{2}=\dfrac{{a}^{2}t}{2}

⇒Wall=12m[(a2t2)2−0]\Rightarrow {W}_{all}=\dfrac{1}{2}m\left[{\left(\dfrac{{a}^{2}t}{2}\right)}^{2}-0\right]

⇒Wall=18ma4t2\Rightarrow {W}_{all}=\dfrac{1}{8}m{a}^{4}{t}^{2}

⇒Wall=∆K\Rightarrow {W}_{all}=∆K

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