Find the constant of motion

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

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