Question

Given in figure above are examples of some potential energy functions in one dimension. The total energy of the particle is indicated by a cross on the ordinate axis. In each case, specify the regions, if any, in which the particle cannot be found for the given energy. Also, indicate the minimum total energy the particle must have in each case. Think of simple physical contexts for which these potential energy shapes are relevant.

Answer 1

Total energy of a system is given by the relation:
$K.E.=E-P.E.$
Kinetic energy of a body is a positive quantity. It cannot be negative. Therefore, the particle will not exist in a region where K.E. becomes negative.
(i) For $x>a,P.E.\left(V_0\right)>E,K.E.$ becomes negative. Hence, the object cannot exist in the region x > a.
(ii) For entire x-axis, $P.E.>E,K.E.$ is negative. Hence, object cannot exist for any x
(iii) For$x<aandx>b,P.E.>E,K.E.$ is negative. Hence, object can exist in region $a<x<b$
(iv) For $-b/2<x<-a/2$ and $a/2<x<b/2,P.E.<E,K.E.$is positive.
Hence, object can exist in this region.