Question #43201

A large inclined plane makes an angle $\theta = 30^\circ$ with the horizontal. A transparent flat glass slab of thickness $d = 3 \text{ cm}$ and refractive index $\mu = \sqrt{3}$ is placed flush on the inclined plane. A small particle is projected from the top surface of the slab with an initial speed $u = 10 \text{ m/s}$ at an angle $\alpha = 30^\circ$ relative to the inclined plane, in the upward direction along the vertical plane containing the line of greatest slope. The particle continuously emits a narrow laser beam along the direction of its instantaneous velocity vector. Find the magnitude of the rate of change of the lateral shift of the laser beam through the glass slab at the exact instant when the particle's velocity becomes purely horizontal. (Take $g = 10 \text{ m/s}^2$. Assume the laser beam always enters the top surface of the slab and exits from the bottom surface).