A projectile problem (Olympiad)

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?

tiny-tim

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.

Michael Si

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?

tiny-tim

Well … either up or down!

"flight" doesn't include landing …