Find the constant of motion

Consider a particle with mass m=3 gm=3~\text{g} that can move on the x-axis and interacts with a wave moving to right with speed u=1000 m/su=1000~ \text{m}/\text{s}. The interaction energy between the particle and the wave is given by V(xut)V(x-u t) where V(x)V(x) is an unknown function. Because the interaction energy is time-dependent the usual total energy of the particle E(t)=mx˙22+V(xut), E(t)= \frac{m \dot{x}^2}{2}+V(x-u t), is not conserved. That is, dE(t)dt0\frac{dE(t)}{dt}\neq 0. Nonetheless, one can show that for certain β\beta the quantity I(t)=mx˙22+V(xut)βx˙ I(t)=\frac{m\dot{x}^2}{2}+V(x-ut)-\beta \dot{x} is conserved, i.e., dI(t)dt=0\frac{dI(t)}{dt}=0. Find β\beta in kg m/s.

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