Power applied by a train engine varies with the time as P = (3t2 + 4t − 1) W where t isin seconds. Calculate the change in kinetic energy of the engine in between 3 s to 5 s.14

The change in kinetic energy of the engine between 3s to 5s can be calculated by integrating the power function over the time interval from 3s to 5s.

The power function is given by P = 3t^2 + 4t - 1.

The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the train engine is equal to the power it applies over a period of time.

So, the change in kinetic energy ΔK.E. = ∫P dt from t=3s to t=5s.

= ∫(3t^2 + 4t - 1) dt from t=3s to t=5s.

= [t^3 + 2t^2 - t] from t=3s to t=5s.

= (5^3 + 25^2 - 5) - (3^3 + 23^2 - 3)

= (125 + 50 - 5) - (27 + 18 - 3)

= 170 - 42

= 128 Joules.

So, the change in kinetic energy of the engine between 3s to 5s is 128 Joules.