Question

The potential energy of a particle of mass 1 kg moving along x-axis given by $U\left(x\right)=[\frac{x^2}{2}-x]J$. If total mechanical energy of particle is 2 J, then find its maximum speed.

Answer 1

Mass of the particle moving along

$x-axis=m=1kg$

The potential energy

$U\left(x\right)=[\frac{x^2}{2}-x]J$

Total mechanical energy of the particle$=2J$

Find maximum speed$=?$

Given,

Kinetic energy$=[2J-(\frac{x^2}{2}-x\left)\right]J$

For maximum speed of Kinetic energy should be maximum

That is ,

$\frac{dK}{dx}=0(\frac{dx}{x}-1)=0x=1K.E=[2-(\frac{1^2}{1}-1\left)\right]J=[2-(\frac{1}{2}-1\left)\right]J=2-\frac{1}{2}=\frac{3}{2}J=1.5JK.E=\frac{1}{2}m{v}^{2}2K.E=m{v}^2{v}^2=\frac{2K.E}{m}v=\sqrt{\frac{2K.E}{m}}=\sqrt{\frac{2\times 1.5}{m}}m/s=\sqrt{\frac{2\times 1.5}{2}}$

(where $m=1kg$)

$=\sqrt{3}=1.732m/s$