Antigravity in the summertime

Summer is coming (for some of us) and you're trying out your new water guns. Your test involves hovering a cafeteria tray in midair by shooting it with a stream of water from your gun. Whichever gun can suspend the tray at the greatest height is the winner.

Suppose your best gun shoots water upward with an initial velocity \(v_0\), and that the weight of your tray is \(M\), what is the height \(h_\text{max}\) (in meters) at which the tray hovers at rest?

Assumptions and Details

  • \(v_0=20\) m/s
  • \(M=1.5\) kg
  • The flux of water from the tip of the gun is \(J_m = 0.45\) kg/s
  • Take \(g = 9.8\) m/s\(^2\)
  • Assume that the water comes straight down after hitting the tray, i.e. the collision is elastic (practically, this can be achieved if the spray has a slight radial spread as it travels upward).

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