Calculate speed of light beam cast onto sphere surface Given an object (B) moving at constant speed and direction directly above a sphere (S) casting a light (L) on the surface of the sphere, how do I calculate the surface speed of the light at any given point or time? See this figure for visualizing the problem If B is traveling at 3500 km/h and the diameter of the sphere is 14000 km, then the speed of the light beam above each point on the diameter line would be constant at 3500 km/h, but on the surface I would expect the speed to be very high on each side of the sphere, but close to 3500 km/h in the middle (top point of sphere). Is there a formula for this?

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Hence $$x=R\cos \theta$$ $$\theta=\cos^{-1} (\frac{x}{R})$$ Now we can get, $$\frac{d\theta}{dt}=\frac{-1}{\sqrt{R^2-x^2}}\frac{dx}{dt}$$ Now using basic knowledge of circular motion, $$v_{\text{surface}}=\color{red}{-}R\frac{d\theta}{dt}=\frac{R}{\sqrt{R^2-x^2}}\frac{dx}{dt}$$ $$v_{\text{surface}}=\frac{R}{\sqrt{R^2-x^2}}v_{object}$$

> We have the $\color{red}{\text{minus}}$ because $\omega$ is anticlockwise, which would give us $v$ in opposite direction of what we want.