Maximizing Free Fall Distance: Solving for Equal Time Intervals

[Juliusz]

Homework Statement

A body is falling from a height of 720m. Divide this distance into three parts so that the body would overcome all parts in the same time.

P.S (This is translated from a different language)

Homework Equations

x(t)= x₀+V₀+at²/2
v(t)=V₀+at

The Attempt at a Solution

I made an acceleration over time graph, and thought about diving it into three equal parts, and the area would be the velocity. But I can't figure out how to divide it into three equal parts, or if this is the correct way to solve the problem at all.
The graph: https://www.desmos.com/calculator/l9nkar1tqa

 

[Doc Al]

What's the total time it takes to fall? Start with that.

 

[Juliusz]

Is this correct? If so, what's next?

x(t)= x₀+V₀+at²/2
t=√2x/√a
t=12.12

Also, how do I format division as a fraction here? Thanks!

 

[Let'sthink]

You can also think of Change in velocity during these equal intervals of time.

 

[PeroK]

Is this correct? If so, what's next?

x(t)= x₀+V₀+at²/2
t=√2x/√a
t=12.12

Also, how do I format division as a fraction here? Thanks!

If you take $g = 10/s^2$ then the time is exactly $12s$.

In any case, now you have the total to time isn't the next step obvious? Divide the motion into three equal parts, it said.

 

[Juliusz]

I looked at the answer, ant it is 80m, 240m, 400m. I can't see how I can figure that out by knowing the total time

 

[Doc Al]

I looked at the answer, ant it is 80m, 240m, 400m. I can't see how I can figure that out by knowing the total time

Divide the total time into thirds (as PeroK told you) and find the position at times t/3 and 2t/3.

 

[Juliusz]

t₁=12
t₂=8
t₃=4

x(t)=a*t²/2

x(t₁)=10*12²/2=720
x(t₂)=10*8²/2=320
x(t₃)=10*4²/2=80

I think I did what you two were suggesting, but I still got a bad answer for t₂. Maybe I made a wrong assumption somewhere?

 

[PeroK]

t₁=12
t₂=8
t₃4

x(t)=a*t²/2

x(t₁)=10*12²/2=720
x(t₂)=10*8²/2=320
x(t₃)=10*4²/2=80

I think I did what you two were suggesting, but I still got a bad answer for t₂. Maybe I made a wrong assumption somewhere?

Think about what those distances mean.

 

[Juliusz]

Think about what those distances mean.

They mean the distance fallen by the body after a certain amount of time

 

[Doc Al]

They mean the distance fallen by the body after a certain amount of time

So how far did the object fall between t₃ and t₂ (for example)?

 

[Let'sthink]

Visualize the problem and the path. Re-read the question. You are just applying the formula blindly and thinking that this formula will give me that answer. You are almost there. Then you are not confident of what you are doing; you say, "I did what you told me" still I am not getting answer. Now just think visualize coolly and alone and get the answer.

 

[Juliusz]

So how far did the object fall between t₃ and t₂ (for example)?

I get it now, thanks! The body fell 80 meters in the first 4 seconds, then another 240 meters (which is 320-80) in the following 4 seconds. And then, in the last 4 seconds it fell 400 meters, which is 720-320m. Thanks for the help everyone!

 

[Doc Al]

Yay!

 

[Let'sthink]

I get it now, thanks! The body fell 80 meters in the first 4 seconds, then another 240 meters (which is 320-80) in the following 4 seconds. And then, in the last 4 seconds it fell 400 meters, which is 720-320m. Thanks for the help everyone!

Good now can you write down a general formula for finding distance traveled between any time t1 and t2, that is for the time interval (t2-t1)? Think it over.