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]

Welcome to PF!

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.

 

[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]

If the rock doesn't fall back, where would it go?

Well … either up or down!

"flight" doesn't include landing …