What Path Should a Mass Take for Shortest Descent Time Under Gravity?

[damnson2000]

Guys I have the following homework problem to solve:
There are 2 given points in a plane. If we take a point-like object with mass m and take it to the "higher" point what path should it go on to reach the other point in the shortest possible time. Only gravitational force affects our point-like object.
My attempt to solution:

Let point A has (x₁, f(x₁)) and point B (x₂, f(x₂)) coordinates.

As we all know average velocity = whole distance/whole time, therefore whole time = whole distance/average velocity.
So we just have to calculate the components of the equation than plug in.

Average velocity:

First we need velocity in the function of position which can be calculated from potential energy difference.

ΔE_potential=ΔE_kinetic
m*g*(f(x₁)-f(x))=1/2*m*v²
so v²=2*g*(f(x₁)-f(x))
therefore v or more likely v(x) =√(2*g*(f(x₁)-f(x))) .

Now we have the velocity in the function of position, all we have to do is to calculate the average value of this function, which is

v_average=∫_x1^x₂√(2*g*(f(x₁)-f(x))) dx / (x₂ - x₁)

Now we need the arc length of our f(x) curve in the intervall of x₁ and x₂.
so
s_whole= ∫_x1^x₂√(1+f²(x)) dx

Now divide the whole distance with average velocity:

t_whole = s_whole/v_average

now we would have to take the quotient of these two expressions and solve its derivate for zero.

∫_x1^x₂√(1+f²(x)) dx * (x₂ - x₁)​

d __________________________________ = 0

∫_x1^x₂√(2*g*(f(x₁)-f(x))) dx

_______
dx
But u haven't managed to find the solution of this equation and I am not even sure if my solution is wether correct or not.
If anyone has an idea how to solve this equation or have found a problem in my solution I am listening with open ears.
​

 

[PeroK]

Is this a question that you made up yourself?

 

[mfb]

The question cannot be solved with the level of tools you use in your post. You'll need calculus of variations.

For the average velocity, you cannot integrate over x like you did, you have to integrate over time.

 

[damnson2000]

Is this a question that you made up yourself?

No, as I said its homework.

 

[damnson2000]

The question cannot be solved with the level of tools you use in your post. You'll need calculus of variations.

For the average velocity, you cannot integrate over x like you did, you have to integrate over time.

Why? I have the velocitys in the function of position, so I integrated that way. Its basically continuous function which has an average value, that's what we need as i know.

 

[mfb]

Why? I have the velocitys in the function of position, so I integrated that way. Its basically continuous function which has an average value, that's what we need as i know.

Travel 100 km at 50 km/h and 100 km at 350 km/h. What is your average velocity?
Your formula would suggest 200 km/h, but then you could travel the whole 200 km in just one hour - in reality the first part alone needs two hours. The total time is 2+(2/7) hours, and the average velocity is 87.5 km/h.

No, as I said its homework.

Did you cover calculus of variations in your class?

Did you copy the exact homework problem word by word?

 

[damnson2000]

Travel 100 km at 50 km/h and 100 km at 350 km/h. What is your average velocity?
Your formula would suggest 200 km/h, but then you could travel the whole 200 km in just one hour - in reality the first part alone needs two hours. The total time is 2+(2/7) hours, and the average velocity is 87.5 km/h.Did you cover calculus of variations in your class?

Did you copy the exact homework problem word by word?

I copied it exactly, but we do not cover variations. Guess I just have to find another way to solve the problem. Thx for the help tho.

 

[Ray Vickson]

I copied it exactly, but we do not cover variations. Guess I just have to find another way to solve the problem. Thx for the help tho.

The curve you seek is called a "brachistochrone". Try a Google search.

 

[damnson2000]

Guys i may have another idea to solve the problem, but in order to make sure my solution is right i need you guys to clarifai the following statements:
If an object moves in one dimension(line) and the speed of the object is given in the function of position the whole time required to the object to finish the distance is equal to:
∫₀^s(dx/v(x))
where s is the total distance that the object going to move, and v(x) is the speed in function of position.

 

[mfb]

That is correct.

 

[damnson2000]

That is correct.

By using the same method which I've used before I've got the following equation:
t= ∫_x1^x₂√((1+f^'(x)^2)/(2g(f(x)-f(x₁)))dx ,
where t is the time required to finish the whole path.
So we need the find the minimal value of this expression with x₁ and x₂ fix parameters, which atm i don't know how should i do, but i'll keep on trying.