A rock is launched vertically. During the last second of the flight, the rock covers one-half of the entire distance covered during the flight. What is the maximum possible duration of the flight? (Hint: answer is not 2 seconds.)

I've tried to use integration to solve an equation of motion which includes air resistance but failed. Anyone can help me?
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 Quote by Michael Si A rock is launched vertically. During the last second of the flight, the rock covers one-half of the entire distance covered during the flight. What is the maximum possible duration of the flight? (Hint: answer is not 2 seconds.) I've tried to use integration to solve an equation of motion which includes air resistance but failed. Anyone can help me?
Hi Michael! Welcome to PF!

i] ignore air resistance (this is a standard uniform-acceleration problem)

ii] you have noticed the question doesn't say the rock returns?

ii] show us what you've tried, and where you're stuck, and then we'll know how to help.
 Thank you. Well, you're right. Since the conditions aren't clearly stated, factors like air-resistance can be ignored so as to simplify the problem. And indeed, the rock would not necessarily return. But I am not sure where the rock would be if i doesn't fall back. If it follows a simple projectile trajectory (in which it falls back to the ground) without resistance and assuming constant acceleration, the answer would be 2 seconds, which isn't the desired answer. Then I considered air drag and formed and solved the differential equation m*dv/dt=mg-Dv^2, which gave me very complicated results. I knew it wasn't the right way then. If the rock doesn't fall back, where would it go?

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