a rain of mass 4000 metric tons is moving with a velocity of 18KMPH. with the application of a force, its velocities changes to 'v' in 2 mins. if the power of the engine is 10MW, the value of 'v' is? (ans : 90)

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Please find below the solution to the asked query:

Given information

mass of the train $m=4000 metric tonns = 4\times {10}^6 kg$; initial velocity of the train $V_i=18 kmph = 5 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ ; power of the engine is $P=10 MW = 10\times {10}^6 W$ and The time period is $t=2 min = 120 sec$. Let the final velocity of the train is $V_f$. Therefore, according to the conservation of energy,

$$\begin{gathered} K.E_f-K.E_i=P t \Rightarrow \frac{1}{2}m\left(V_f^2-V_i^2\right)=10\times {10}^6\times 120 \\ \Rightarrow \frac{1}{2}\times 4\times {10}^6\left(V_f^2-5^2\right)=1200\times {10}^6 \Rightarrow V_f^2-25=600 \Rightarrow V_f^2=600+25 \\ \Rightarrow V_f^2={25}^2 \Rightarrow V_f=25 \raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \Rightarrow V_f=25\times \frac{18}{5} \\ \Rightarrow V_f=90 kmph \end{gathered}$$

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