Is anyone familiar with this problem? HELP!!?

Question

If you could dig a straight shallow tunnel through the earth, say from NY to LA, and then evacuate it and put in a zero friction track, you could roll a train from one end to the other with no propulsion system other than gravity. As the train left NY it would roll very slowly, gain speed until it got half way to LA, then start to slow down, pull very gently into the LA station, then stop.

I need mainly the time it takes for the train to go from NY to LA.
I need to know the calculations needed and how to do them.
Thanx!

Answer 1

The previous suggestion of calculating the potential energy is good - I started doing it that way but realized this was easier:

Let x be the distance along the tunnel from its center. The acceleration of gravity anywhere in the earth is gr/R pointing radially downward where r is the distance to the center, R is the earth radius and g the acceleration of gravity at the surface.

To get the component of this force along the tunnel it has to be multiplied by the cosine of the angle of the tunnel with the straight downward (radial) direction which is just x/r. In other words, the acceleration is just:

a = -gx/R

This has the same form as the force for a simple harmonic oscillator with spring constant k=g/R and unit mass and that is known to have period:

T = 2pi/sqrt(k) = 2pi sqrt(R/g)

The trip from one endpoint to the other takes one half a period so this time is:

t = pi sqrt(R/g)

R is about 6400km and g = 9.8 m/sec so this seems to give about 42 minutes. Amazingly enough it seems to be the same for any straight line path connecting two points on the surface.

Answer 2

Your acceleration will be based on how deep the tunnel is. The gravity propels the train so potential energy is the only thing that will make the train move. calculate the potential energy and you can use that figure to back out velocity and then time.

Answer 3

Mrs. Payne is that you? I thought I get rid of these math questions when I graduated.