If a variable force of magnitude F(x) moves an object of mass m along the x-axis from x₁ to x₂, the object’s velocity y can be written as dx/dt (where t represents time). Use Newton’s second law of motion F = m(dy/dt) and the Chain Rule

to show that the net work done by the force in moving the object from x₁ to x₂ is

where ν₁ and ν₂ are the object’s velocities at x₁ and x₂. In physics, the expression (1/2)mν² is called the kinetic energy of an object of mass m moving with velocity ν. Therefore, the work done by the force equals the change in the object’s kinetic energy, and we can find the work by calculating this change.

du dt dv dx dx dt - U du dx