Why is work done force times displacement?

Question

Why is work done the product of force and displacement? Why not force and time?

Answer 1

If work is done, there must be a change in the kinetic energy of the system. Therefore, a force needs to be applied on an object over a distance to either increase or decrease its kinetic energy. That is $$W=\vec {F\ }\cdot \vec {ds}=\Delta \text{kinetic energy}$$

The definition of impulse is force $\times$ time. This however is equal to the objects change in momentum since $$\vec F\Delta t=m\Delta \vec v =\Delta \text{momentum}$$ which certainly is not equal to a change in kinetic energy.

So we define work done as force $\times$ distance as it is identical to the change in kinetic energy, but impulse we define as force $\times$ time which is identical to its change in momentum.

Also, it is worthwhile noting that work done, which is a scalar quantity, would have to be a vector quantity if we were to say that work is force $\times$ time, since force itself is a vector while time is a scalar. The multiplication of a vector by a scalar is still a vector.

Answer 2

Energy transfer always involves an intensive quantity and a conjugate extensive quantity (that multiply to give units of energy). A gradient—meaning a spatial difference—in the intensive parameter drives the process, and shifting of the extensive parameter transfers the energy.

For mechanical work, force is the intensive quantity, and distance is the extensive quantity. For compressive work: pressure/volume. For elastic deformation work: stress/volumetric strain. For electrical work: field/charge. For surface energy work: surface tension/area. For heat transfer: temperature/entropy. For chemical reactions, phase changes, and diffusion: chemical potential/matter. The framework underlies all of thermodynamics.