What is the difference between Kinetic energy and Work?

Question

Can we say that the work is the difference in the energy of motion

Answer 1

We can say that work $W$ done by a force $\vec{F}$ acting on an object is equal to the change in the object's kinetic energy $\Delta{K}$ during the time the force acts. This is the work-energy theorem. The proof for a constant force is simple:

$$a = F/m,$$

where $a$ is constant since $F$ is constant, and $m$ is the object's mass. Since $a$ is constant, we can use one of the kinematic equations to write

$$a = F/m = \frac{v_{f}^{2} - v_{i}^{2}}{2\Delta{x}},$$

where $v_i$ is the initial speed (when the force begins to act) and $v_f$ is the final speed (when the force stops acting) and $\Delta{x}$ is the object's displacement during the time the force is acting.

The work is calculated from $W = F\Delta{x}$ assuming the force acts entirely in the direction of the displacement. Otherwise, use the component of the force acting along the line of $\Delta{x}$. (You have to use that component to get $a$ too.)

Now, substitute for $F$:

$$W = F\Delta{x} = \frac{v_{f}^{2} - v_{i}^{2}}{2\Delta{x}}m\Delta{x}$$ $$W = \frac{v_{f}^{2} - v_{i}^{2}}{2}m = \frac{1}{2}mv_{f}^{2} - \frac{1}{2}mv_{i}^{2} = \Delta{K}.$$

The work $W$ will be positive if $v_f > v_i$ and negative otherwise.

The proof for variable forces is a bit more involved, but we reach the same conclusion: $W = \Delta{K}$.