50 lb Package Sliding Down Chute

A package chute has been installed at a distribution center to quickly move packages from the second floor to the first floor. The center's manager would like to know where a person should stand so that a relatively heavy 50 lb package will stop within reach after bounding off the spring stop.

What is known:

• chute height, 10 ft
• chute angle, 26.6°
• horizontal chute length, 7 ft
• spring constant, k, 50 lb/ft
• package weight, 50 lb
• initial velocity, 0.0 ft/s
• coefficient of friction, μ, 0.30
• package width, 2 ft

If the package is released from rest, where will the package stop? Does the package weight matter?

• Assume the package is a particle and all forces act at its center.
• Assume smooth transition between the sloped and horizontal sections of the chute.
• Use the Principle of Work and Energy:
      T₁ + ΣU_1-2 = T₂
  
• Include friction forces between the chute and package.