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

work = change in energy = integral of P dt

= t^3 - t^2 + t at 4 - at 1

To find the change in kinetic energy between t=1s and t=4s, we need to integrate the power function with respect to time from t=1s to t=4s.

The change in kinetic energy is given by the integral of power with respect to time:
ΔKE = ∫(P dt)

Here, the power function is P = (3t^2 - 2t + 1) W.

Replacing P in the integral:
ΔKE = ∫((3t^2 - 2t + 1) dt)

Integrating each term separately:
ΔKE = ∫(3t^2 dt) - ∫(2t dt) + ∫(1 dt)

∫(3t^2 dt) = t^3 + C1, where C1 is the constant of integration.
∫(2t dt) = t^2 + C2, where C2 is the constant of integration.
∫(1 dt) = t + C3, where C3 is the constant of integration.

Substituting these integrals back into the equation:
ΔKE = (t^3 + C1) - (t^2 + C2) + (t + C3)

Now, we evaluate the expression at the upper limit t=4s and subtract it with the evaluation at the lower limit t=1s to find the change in kinetic energy:

ΔKE = [(4^3 + C1) - (4^2 + C2) + (4 + C3)] - [(1^3 + C1) - (1^2 + C2) + (1 + C3)]

Simplifying this expression:
ΔKE = [64 + C1 - 16 - C2 + 4 + C3] - [1 + C1 - 1 - C2 + 1 + C3]

Many of the constants cancel out:
ΔKE = (66 + C1 - C2 - C1 + C3) - (0)

Finally, we obtain the change in kinetic energy:
ΔKE = 66 - C2 + C3

Since the constants of integration (C1, C2, C3) are arbitrary, we cannot determine their exact values. Therefore, the change in kinetic energy between t=1s and t=4s is given by 66 - C2 + C3.