Conservation of energy Suppose an object with mass m moves

QUESTION:

Suppose an object with mass m moves in a conservative force field given by \(\mathbf{F}=-\nabla \varphi\), where \(\varphi\) is a potential function in a region R. The motion of the object is governed by Newton’s Second Law of Motion, \(\mathbf{F}=m \mathbf{a}\), where a is the acceleration. Suppose the object moves (either in the plane or in space) from point A to point B in R.

a. Show that the equation of motion is \(m \frac{d \mathbf{v}}{d t}=-\nabla \varphi\).

b. Show that \(\frac{d \mathbf{v}}{d t} \cdot \mathbf{v}=\frac{1}{2} \frac{d}{d t}(\mathbf{v} \cdot \mathbf{v})\).

c. Take the dot product of both sides of the equation in part (a) with \(\mathbf{v}(\mathrm{t})=\mathbf{r}^{\prime}(\mathrm{t})\) 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 plsu potential) \(\frac{1}{2} m|\mathbf{v}|^{2}+\varphi\) is the same at A and B. Conclude that because A and B are arbitrary, energy is conserved on R.