Viscous Trajectory

This problem was asked to a friend of mine in the KVPY interview. He was asked to describe the motion qualitatively. I wanted to analyze it quantitatively.

A solid wooden sphere with density less than water is released with a speed v in water, along the positive \(x\) axis, at the origin.

Find its maximum \(x\) coordinate (in meters)

Details and assumptions

  • Stoke's law is applicable.
  • The tank is very wide and very deep, so that the sphere loses all horizontal velocity before breaching the surface.
  • Mass of the sphere = \(\SI{6}{\kilo\gram}\).
  • Initial speed = \(\SI{\pi}{\meter/\second}\).
  • Radius of sphere = \(\SI{0.5}{\meter}\).
  • Coefficient of viscosity = 1.
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