Question

Power applied to a particle varies with time as P=(3t²-2t+1) Watt, where t is second. Find the change in its kinetic energy between time t=2s and t=4s
a)32 J
b)46J
c)61J
d)102J

Answer 1

Also dw=(3t²-2t+1)dt
=>w=[t³-t²+t](2s to 4s)
so W=56-12+2=46 J.
It is the kinetic energy.

Answer 2

The change in kinetic energy is b) 46 J

Given:

$P=\left(3 t^{2}-2 t+1\right)$

To find:

Kinetic energy change = ?

Solution:

The formula of power is given below:

$P=\frac{d w}{d t}$

$\Rightarrow d w=P . d t$

Thus, the work done is given as,

$\int d w=\int_{t_{1}}^{t_{2}} P . d t$

The kinetic energy change is equal to work done.

$w=\int_{2}^{4}\left(3 t^{2}-2 t+1\right) d t$

On separating, the integration, we get,

$w=\int_{2}^{4} 3 t^{2} \cdot d t-\int_{2}^{4} 2 t \cdot d t+\int_{2}^{4} d t$

On integrating, we get,

$w=3\left[\frac{t^{3}}{3}\right]_{2}^{4}-2\left[\frac{t^{2}}{2}\right]_{2}^{4}+[t]_{2}^{4}$

$w=\left[t^{3}\right]_{2}^{4}-\left[t^{2}\right]_{2}^{4}+[t]_{2}^{4}$

On substituting the value, we get,

$w=[64-8]-[16-4]+[4-2]$

$w=56-12+2$

$\Rightarrow w=58-12$

Thus, the change in kinetic energy is,

$\therefore w=46 \ \mathrm{J}$