If I change the parametrization of a line integral would the power in the work change?

Question

Working with line integrals I noticed when the parametrization is changed the answers are the same amount of work. But isn't t the time element as well so if I follow the same path using a different parametrization would not the power of the work change ?

Certainly the vector field stays the same and the path remains constant through the field so if you change t then would not the unit of time change necessarily ?

Answer 1

The power is the work made for unit of time thus there is not contradiction for the fact that work doesn't change with different parametrization.

Indeed we have for a given parametrization $\vec r(t)$

$$W=\int_{t_0}^{t_1}\vec F(t)\cdot \vec r'(t)dt$$

but $t$ here is a parameter which is not related to the time.

Answer 2

Both the work and the power are defined with geometric vectors, so they can't depend on the parametrization.