Question 705021

SCQMEDIUM

The evolution of the dynamical variables x(t)x(t) and p(t)p(t) is given by

x˙=ax\dot{x} = ax

p˙=−p\dot{p} = -p

where aa is a constant. The trajectory in (x,p)(x, p) space for −1<a<0-1 < a < 0 is best described by

(A)
Option A
(B)
Option B
(C)
Option C
(D)
Option D

Detailed Solution

We have dx/dt=axdx/dt = ax and dp/dt=−pdp/dt = -p.

Dividing the two equations: dx/dp=(ax)/(−p)  ⟹  dx/x=−a(dp/p)dx/dp = (ax)/(-p) \implies dx/x = -a(dp/p).

Integrating both sides: ln⁡x=−aln⁡p+C\ln x = -a \ln p + C.

This implies x=kp−ax = k p^{-a}.

Since −1<a<0-1 < a < 0, let a=−0.5a = -0.5, then x=kp0.5x = k p^{0.5}, which is a parabolic shape characteristic of option 1.

Free Exam

Boost Your Exam Preparation!

Move beyond just reading solutions. Access our comprehensive Test Series, original Mock Tests, and interactive learning modules. Many premium tests are completely free!

  • Original Mocks & Regular Test Series
  • Real NTA-like Interface with Analytics
  • Many Free Tests Available