Moms make the best lifeguards

A family decides to go to the lake on their summer vacation. Their child is small, just around 4 or 5 years old and is very excited. When they get to the lake, the child runs immediately into the water while his parents set up their blanket on a little beach by the lake. Out in the lake, the child steps on a rock, hurts his toe, and begins to cry for his mom to come get him.

Mom's velocity is \(v_s=3~m/s\) on the sand and \(v_w=1~m/s\) in the water. The distance between the mom and the child is given in the figure, with \(d=30~m\), \(a=10~m\) and \(b=30~m\). What's the minimum time in seconds for her to reach her child?

Details and assumptions

  • Remember, the quickest path between 2 points on a plane is a straight line (if you travel at constant velocity).
  • There are a couple ways to do this problem, but one straightforward way involves solving a quartic equation, which you can do here.

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