# 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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