G7.) (5 pts) Suppose that a particle with mass m is moving in \mathbb{R} ^ 3 under the influence of a force field vec F (x,y,z) . Assume that the force field vec F has the following special property: overline F =- nabla V , where V(x, y, z) is some smooth scalar- valued function (here, V / (\mathbb{R} ^ 3) -> \mathbb{R} and VV is the gradient of V) . vec F Denote by vec r (t) the position of the particle at time t. Assume that the usual Second Newton's Law is applicable to our situation: m vec a = vec F where stands for the acceleration vector of the particle (remember vec a the formula : vec a =( vec r )^ prime prime ) . Define the total energy function E(t) of the particle at time t as: E(t)=(V of r )+ (m||( vec r )^ prime ||^ 2)/2 , where (V vec r ) is the composition of V(x, y, z) and vec r (t) (Note that vec r depends on t, so that the right hand side is a function of t. We suppress t in our writing of vec r (t) and simply vec r to make the formulas look simpler .) ( i) Explain briefly why the second term on the right hand side of the formula for E(t) can be branded as the " kinetic energy" of the particle at time t. (Hint: Recall the physical meaning of ( vec r )^ prime as the velocity of a particle and its magnitude ||( vec r )^ prime || as the speed of the particle.) ( ii) Denote by K(t) the second term on the right hand side right hand side of the formula for E(t) Show that K^ prime (t)=m( vec r )^ prime prime *( vec r )^ prime = vec F *( vec r )^ prime ( Hint: To prove the first equality, use one of the properties of the dot product to rewrite ||( vec r )^ prime ||^ 2 =( vec r )^ prime *( vec r )^ prime . use the product rule for p