# Physics ( Calculus & Mecahnics )

A proton with mass m moves in one dimension.The potential-energy function is where and are positive constants. The proton is released from rest a x0 = a/b.
(a) Show that can be written as

U(x) = a
--- [ ( x0/x )^2 - x0/x ]
x0^2

Graph U(x). Calculate U(x0) and thereby locate the point on
the graph. (b) Calculate v(x) the speed of the proton as a function
of position. Graph 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 = 3a/B 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?

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