Chapter 14.3, Problem 52E

Conservation of energy Suppose an object with mass m moves in a region R in a conservative force field given by F = –▿φ, where φ is a potential function in a region R. The motion of the object is governed by Newton’s Second Law of Motion, F = ma, where a is the acceleration. Suppose the object moves from point A to point B in R.

a.    Show that the equation of motion is $m\frac{dv}{dt}=-\nabla \phi$.

b.    Show that $\frac{dv}{dt}·v=\frac{1}{2}\frac{d}{dt}(v·v)$.

c.    Take the dot product of both sides of the equation in part (a) with v(t) = r′(t) and integrate along a curve between A and B. Use part (b) and the fact that F is conservative to show that the total energy (kinetic plus potential) $\frac{1}{2}m{\left|v\right|}^2+\phi$ is the same at A and B. Conclude that because A and B are arbitrary, energy is conserved in R.