greypilgrim
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Hi.
I'm trying to derive Snell's law from a mechanical model: An axle of length ##d## with two wheels that run at speeds ##v_1## or ##v_2##, depending on the medium they are in (##v_1>v_2## for the moment). The ##x##-axis is the interface, ##v_1## for ##y>0## and ##v_2## for ##y<0##. The wheels come in at an angle ##\alpha_1## from the top left and the origin is where the first wheel (the right one in direction of motion) enters the slow medium. Here's what I did:
Is there a better way to do this, or maybe a cleverer choice of the coordinate system?
I'm trying to derive Snell's law from a mechanical model: An axle of length ##d## with two wheels that run at speeds ##v_1## or ##v_2##, depending on the medium they are in (##v_1>v_2## for the moment). The ##x##-axis is the interface, ##v_1## for ##y>0## and ##v_2## for ##y<0##. The wheels come in at an angle ##\alpha_1## from the top left and the origin is where the first wheel (the right one in direction of motion) enters the slow medium. Here's what I did:
- Calculated the turn radius (measured to the outer, i.e. left, wheel) ##r## using proportionality with ##d##, ##v_1## and ##v_2##.
- Calculated the centre of the turn using trigonometry with ##(r-d)## and ##\alpha_1##.
- The turn is over when the left wheel enters the slow medium. Hence I intercepted the turn circle of radius ##r## with the ##x##-axis and took the positive solution.
- Calculated the exit angle ##\alpha_2##.
Is there a better way to do this, or maybe a cleverer choice of the coordinate system?