Question

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

• A. $32J$
• B. $46J$
• C. $61J$
• D. $100J$

Answer 1

Answer: (B)

$P=3t^{2}-2t+1=\frac{d E}{d t}$
$\therefore \, dE=\left(\right.3t^{2}-2t+1\left.\right)dt$
$E=\displaystyle \int _{t = 2 s}^{t = 4 s} \left(\right. 3 t^{2} - 2 t + 1 \left.\right) d t$
$=\left[\frac{3 t^{3}}{3} - \frac{2 t^{2}}{2} + t\right]_{t = 2 s}^{t = 4 s}$
$=\left[\right.\left(4^{3} - 2^{3}\right)-\left(4^{2} - 2^{2}\right)+\left(\right.4-2\left.\right)\left]\right.$
$E=56-12+2=46 \, J$