What is Work and how is it related to Kinetic Energy

• Work $W$ is the change in the kinetic energy of a body due to an external force.

$$\begin{array}{r}W=K_f-K_i\\ \\ W=\vec{F}\cdot \vec{s}\end{array}$$

• It accounts for the increase (or decrease) of velocity; i.e. transfer of energy.
  • Increase in object’s velocity → Force has done positive work on the body.
  • Decrease in object’s velocity → Force has done negative work on the body.
• It’s in Joules since it’s just change in energy (which is in Joules).

Alternate expression of work

If our object goes from $v_0$ to $v$ in x-axis:

$$\begin{array}{r}W=\frac{m{v}^2}{2}-\frac{m{v}_0^2}{2}\\ \\ =\frac{m}{2}(v^2-v_0^2)\\ \\ =\frac{m}{\cancel{2}}(\cancel{2}ax)\\ \\ =(ma)x\\ \\ W=F_{x}x\\ \\ \text{Generalizing for different axes:}\\ \\ =(\vec{F}\cos \theta )\vec{s}\\ \\ W=\vec{F}\cdot \vec{s}\end{array}$$

• Since we need the component of force only along the axis of motion, we use $\cos \theta$.