Question 15.3.27

Suppose you’re standing to the south of a lake, with the bank running straight east-west. You want to run to the water’s edge and swim to a boat anchored in the lake, some distance to the north-east. You can run faster than you can swim. Your running speed is $v_1$ and your swimming speed is $v_2$. Diagram: snell 1

  1. If you want to get to the boat as quickly as possible, what do you think your path should look like? Show answer

  2. Show that \[\frac{\sin\theta_1}{v_1} = \frac{\sin\theta_2}{v_2}\] with $\theta_1$ and $\theta_2$ defined as in the previous answer. Some of you may recognise this as Snell’s law, which describes the path that light takes when travelling from air to glass or water, for example (refraction). This is because light travels more slowly in denser materials and likes to take the quickest path (Fermat’s principle). Show hint

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