Sphere with Tunnel Homework: Shortest Path Solution

[nfm1337]

Homework Statement

A train slides without friction inside a tunnel drilled through Earth from
station A to station B. Find the curve the tunnel should follow, in order for the duration of traveling from A to B to be as short as possible. Assume Earth to be a (non-rotating) homogeneous sphere.

Homework Equations

Transit time:
$$T_{AB} = \int_A^B \frac{ds}{V}$$

Euler-Lagrange:
$$\frac{\delta f}{\delta y} - \frac{d}{dx} \frac{\delta f}{\delta f'} = 0$$

The gravitational acceleration at a distance r from the centre of the sphere, if r < the radius of the sphere:
$$g = \frac{4}{3} \pi \rho G r$$

The Attempt at a Solution

I tried to find an expression for the velocity, v. First by using the law of conservation of energy $$1/2 mv^2 = mgh$$, but then when I plugged this expression into the integral for the transit time, and then plugged the expression to be integrated into the Euler-Lagrange equation, I ended up with differential equations which seemed impossible to me. Then by trying to find an expression for the acceleration, as the scalar product of the gravitational acceleration and the unit tangent vector of the curve. But again, I ended up with horrible differential equations.

Then I checked on MathWorld (http://mathworld.wolfram.com/SpherewithTunnel.html" ) but I couldn't even understand how they reached the first equation.

I would be very happy if someone could help me with this problem. I've spent all day on it and I feel like an idiot.

 

[anathema]

Thank you for your post. Finding the curve for the tunnel to minimize the transit time from station A to station B is indeed a challenging problem. However, with the equations you have provided, I believe we can work towards a solution together.

First, let us define some variables to make our equations more manageable. Let r be the distance from the center of the Earth to any point on the tunnel, and let θ be the angle between the tangent to the curve and the radial direction at that point. We can also define v as the speed of the train at any point on the curve.

Next, we can use the law of conservation of energy to find an expression for v. Since there is no friction, the only force acting on the train is the gravitational force. Therefore, we can write:

1/2 mv^2 = mgh

Where m is the mass of the train, g is the gravitational acceleration at a distance r from the center of the Earth, and h is the height of the train above the surface of the Earth. We can express h in terms of r and θ using basic trigonometry:

h = r(1-cosθ)

Substituting this into the equation for conservation of energy, we get:

1/2 mv^2 = mgr(1-cosθ)

Solving for v, we get:

v = √(2gr(1-cosθ))

Now, we can use this expression for v in the transit time integral:

T_{AB} = ∫_A^B ds/v

Since we are trying to minimize the transit time, we can treat it as a functional and use the Euler-Lagrange equation to find the curve that minimizes it. The Euler-Lagrange equation for this problem can be written as:

∂/∂θ (∂f/∂v) - d/dθ(∂f/∂v') = 0

Where f is the integrand in our transit time integral, and v' is the derivative of v with respect to θ. Plugging in our expression for v and simplifying, we get:

∂/∂θ (1/v) - d/dθ(v/v') = 0

Using the chain rule and simplifying, we get:

d/dθ(v^2) = 2v(v')^2 + v^3

Now, we can substitute in our expression

 

[nlightner]

First of all, don't feel like an idiot. This is a challenging problem and it's completely normal to struggle with it. It's great that you tried different approaches and consulted outside sources for help.

To solve this problem, we need to find the curve (or path) that minimizes the transit time from station A to station B. This is known as the brachistochrone curve, and it can be found using the calculus of variations, which is what the Euler-Lagrange equation is for.

The first step is to define the problem in mathematical terms. Let's say the curve of the tunnel is described by the function y(x), where x is the distance along the tunnel and y is the height of the tunnel at that point. We also need to define the transit time T as a function of x and y, which we can do using the given equation:

T(x,y) = \int_{A}^{B} \frac{ds}{V} = \int_{A}^{B} \frac{\sqrt{1+y'^2}}{\sqrt{2gy}} dx

where y' = dy/dx and g is the gravitational acceleration given in the problem.

Next, we need to apply the Euler-Lagrange equation to this functional. The Euler-Lagrange equation tells us that the function y(x) that minimizes the functional T must satisfy the following equation:

\frac{\partial T}{\partial y} - \frac{d}{dx} \frac{\partial T}{\partial y'} = 0

Plugging in the expression for T and simplifying, we get the following differential equation:

2gy - (1+y'^2)\frac{d}{dx}\left(\frac{y'}{\sqrt{1+y'^2}}\right) = 0

This is a second-order nonlinear differential equation, which can be solved using various techniques such as separation of variables or substitution.

If you're having trouble solving this equation, I suggest consulting a math textbook or online resources for help with solving second-order nonlinear differential equations. Once you have the solution for y(x), you can use it to find the curve of the tunnel that minimizes the transit time.

I hope this helps and good luck with your homework!