power applied to a particle varies with times as p=3t^2-2t+1 Watt where t is in second. And the change in its kinetic energy b/w time t=2s t=4s

p(t)=3t²-2t+1 W

Δp
=p(4)-p(2)
=41-9
=32 W

To find the change in kinetic energy, we first need to calculate the initial and final kinetic energies at times t=2s and t=4s.

The power applied to the particle can be used to calculate the rate of change of its kinetic energy using the formula:

Power = Rate of Change of Kinetic Energy

Given that the power applied to the particle is given by the equation p = 3t^2 - 2t + 1 Watt, we can integrate it to obtain the change in kinetic energy.

∫(3t^2 - 2t + 1) dt = ∫dKE

Integrating the equation, we get:

KE = t^3 - t^2 + t + C

Now, we can calculate the initial and final kinetic energies at t=2s and t=4s.

Initial kinetic energy (KE1) at t=2s:
KE1 = (2^3 - 2^2 + 2) + C = 8 - 4 + 2 + C = 6 + C

Final kinetic energy (KE2) at t=4s:
KE2 = (4^3 - 4^2 + 4) + C = 64 - 16 + 4 + C = 52 + C

The change in kinetic energy (ΔKE) between t=2s and t=4s can be calculated by subtracting the initial kinetic energy from the final kinetic energy:

ΔKE = KE2 - KE1 = (52 + C) - (6 + C) = 52 - 6 = 46 J (Joules)

Therefore, the change in kinetic energy between t=2s and t=4s is 46 Joules.