# Work-kinetic energy theorem for time-dependent force

Does the work-kinetic energy theorem hold in all the cases? Specifically is true for a time-dependent force?

Yes, it still applies for time-dependent forces.

When the force varies with time, you just have to be careful with how you calculate the work.

For example, the equation $$W = F \cdot s$$ only applies if F is constant over the entire path of travel.

If $$F$$ is not constant, we can instead say the instantaneous work at any time is $$\delta W = F \cdot ds$$; and by taking the integral, we can add up the work at all those instantaneous times to find the total work over some set period of time (or path of travel) where the force varies.

If you're not familiar with calculus/integrals, you're essentially calculating a new work at each point in time due to the change in force, and adding all those together to find a total.

Specifically is true for a time dependent force.

Yes.

The work energy theorem states that the change in kinetic energy of an object is equal to the net work done on the object. If the force is time dependent you can apply the theorem for a given time interval by using the average force over the time interval, or the average force over the distance the force is applied, to calculate the average net work done on the object. This is typically done in applying the theorem to inelastic collisions (e.g., car crashes) as

$$F_{ave}d=-\frac {mv_{i}^2}{2}$$

Where $$d$$ is the stopping distance, and $$i$$ is the initial kinetic energy.

Hope this helps.