Speed of a sphere fired straight up, including quadratic air resistance

AI Thread Summary
The discussion revolves around calculating the speed of a sphere fired straight up, factoring in quadratic air resistance while ignoring linear resistance. Participants emphasize that the velocity cannot be modeled using constant acceleration equations due to the variable nature of acceleration influenced by air resistance. Instead, it's suggested to express acceleration as a function of velocity, acknowledging that both upward and downward motions require different considerations. The importance of correctly applying the forces acting on the sphere is highlighted to derive the appropriate equations. Overall, understanding the dynamics of the problem is crucial for accurately determining the sphere's speed upon impact.
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Homework Statement


A sphere of radius 0.2 cm is fired straight up with speed v0. What is its speed when it hits the ground? (Include quadratic air resistance but ignore linear air resistance.)

Homework Equations


Quadratic air resistance: FR = cv2
c = 0.5CpA = 4.05 x 10-6

The Attempt at a Solution


I figured as the sphere rises, its velocity can be modeled as v = v0 - gt - cv2t/m. As it falls, velocity can be modeled as v = gt - cv2t/m. Other than that, I'm not really sure where to go.
 
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Hi soundbyte, welcome to PF

I don't think you can write the velocity like that, as the equation

v = v0 + a.t is for constant acceleration only,

in this problem a varies with v and v=v(t), so a= a(t)...(ie a is a function of t)

A good place to start would be to write the acceleration on the sphere as a function of v based on the forces
 

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