Chapter 7, Problem 7.80CP

CALC A proton with mass m moves in one dimension. The potential-energy function is U(x) = (α/x²) − (β/x), where α and β are positive constants. The proton is released from rest at x₀ = α/β. (a) Show that U(x) can be written as

$U\left(x\right)=\frac{\alpha }{x_0^2}\left[{\left(\frac{x_0}{x}\right)}^2-\frac{x_0}{x}\right]$

Graph U(x). Calculate U(x₀) and thereby locate the point x₀ on the graph. (b) Calculate v(x), the speed of the proton as a function of position. Graph v(x) and give a qualitative description of the motion. (c) For what value of x is the speed of the proton a maximum? What is the value of that maximum speed? (d) What is the force on the proton at the point in part (c)? (e) Let the proton be released instead at x₁ = 3α/β. Locate the point on the graph of U(x). Calculate v(x) and give a qualitative description of the motion. (f) For each release point (x = x₀ and x = x₁), what are the maximum and minimum values of x reached during the motion?