A particle of mass m at position x(t)e, is moving under the effect of a conservative force with corresponding potential V (x). I. (a) Give an expression for the total energy E of the particle, and hence determine an expression for dæ/dt in terms of E, m and V. (b) Show that the force F = -xe-r²e, is conservative, and find the corresponding potential V (x), assuming that V → 0 as x ±x. (c) Show that the motion is bounded for –1/2 < E < 0, and specify the bounds. What happens when E < –1/2, and when E > 0?

A particle of mass m at position x(t)e, is moving under the effect of a conservative force with corresponding potential V (x). I. (a) Give an expression for the total energy E of the particle, and hence determine an expression for dæ/dt in terms of E, m and V. (b) Show that the force F = -xe-r²e, is conservative, and find the corresponding potential V (x), assuming that V → 0 as x ±x. (c) Show that the motion is bounded for –1/2 < E < 0, and specify the bounds. What happens when E < –1/2, and when E > 0?