Death defying skateboard helicopter stall

Classical Mechanics Level 5

Pro skateboarder Bob Burnquist has gigantic ramps in his backyard, and he's always trying to do bigger and better things with them. Recently, Bob got a pilot to hover over his gigantic quarterpipe so that he could air and stall on the skids of the helicopter. Due to gravity alone, Bob has to go pretty fast to air out of the quarterpipe at all. However, when the helicopter is hovering above the ramp, it provides a massive downward thrust which means that Bob must go even faster. This is a terrifying experience because as you approach the ramp you feel as if you have enough speed to go straight through the blades. The crucial factor is that the air being pushed through the blades works to decelerate Bob's ascent.

What is the minimum speed (in m/s) Bob must have at the top of the ramp in order to stall on the skids of the copter hovering 5 m above the ramp?

Assumptions and Details

  • The density of air is 1 kg/m\(^3\)
  • \(g\) = 9.8 m/s\(^2\)
  • The mass of the helicopter \(m_H\) is 2500 kg
  • The mass of Bob \(m_\textrm{Bob}\) is 70 kg
  • The circle swept out by the rotating blades has radius \(r_b=\) 4 m
  • Bob has a cross sectional area of \(A_\textrm{Bob}=\) 0.4 m\(^2\)
  • The helicopter hovers by accelerating a cylindrical column of air downward, and Bob is always within this cylinder.
  • Bob has a drag coefficient of 1

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