Question

Power applied to a particle varies with time as $ P=(3{{t}^{2}}-2t+1) $ W, where t is in second. Find the change in its kinetic energy between t = 1 s and t = 4 s.

• A. 32 J
• B. 46 J
• C. 61 J
• D. 102 J

Answer 1

Answer: (B)

The average power or simply power is the average amount of work done or energy transferred per unit time. The instantaneous power is then the limiting value of the average power as the time interval $ \Delta t $ approaches zero.
$ P=\underset{\Delta t\to 0}{\mathop{\lim }}\,\frac{\Delta W}{\Delta t} $ $ \therefore $ $ W=\int{{}}Pdt $
Given, $ P=3{{t}^{2}}-2t+1 $
$ \therefore $ $ W=\int_{2}^{4}{(3{{t}^{2}}-2t+1)dt} $
Using $ \int{{}}{{x}^{n}}dx=\frac{{{x}^{n+1}}}{n+1} $ ,
we have $ W=[{{t}^{3}}-{{t}^{2}}+t]_{2}^{4}=56-12+2 $
$ \Rightarrow $ $ W=46\,\,J $