CALC? A proton with mass m moves in one dimension. The potential-energy function is U(x) = ?/x2) – (?/x), where ? and ? are positive constants. The proton is released from rest at x0 = ?/?. (a) Show that U(x) can be written as Graph U(x). Calculate U(x0) and thereby locate the point x 0 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 pro-ton be released instead at x1 = 3?/?. Locate the point x1 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?

Solution 87CP Step 1: 2 The potential energy fraction is U(x) = /x (/x). Where and are positive constants. The proton is released from rest at x = / . 0 2 2 U(x) = /x 0(x /x0 (x /x)0……..(1) 2 By factoring /x 0 on R.H.S of eq……..(1).where = /x 0 U(x) = /x .x /x /x x 0 0 0 U(x) = /x 0x /x0 /x 02 x0/x 2 2 U(x) = /x (0 /x 0 x /x) 0 The curves of U(x) and V (x) are shown in above graphs. U(x) = 0,while U(x) is positive for x < x and U(x) is negative for x > x . 0 0