Gravity Train 3

Classical Mechanics Level 5

Two cities are situated at an angular distance of θ \theta with respect to Earth's center.

You are asked to find a broken-line tunnel A C B ‾ \overline{ACB} through Earth’s crust connecting the two cities such that it minimizes the time T T to commute between the cities when the train moves only under the influence of gravity.

What is this minimum T T ?

Details and Assumptions:

  • Assume Earth to be a sphere of uniform mass density with radius R R .
  • Ignore friction and collision.
2 R g sin ⁡ − 1 ( 1 − sin ⁡ θ 2 1 + sin ⁡ θ 2 ) 2\sqrt{\frac Rg}\sin^{-1}\left(\frac {1-\sin\frac\theta 2}{1+\sin\frac\theta 2}\right) 2 R g sin ⁡ − 1 ( 1 − cos ⁡ θ 2 1 + cos ⁡ θ 2 ) 2\sqrt{\frac Rg}\sin^{-1}\left(\frac {1-\cos\frac\theta 2}{1+\cos\frac\theta 2}\right) 2 R g cos ⁡ − 1 ( 1 − cos ⁡ θ 2 1 + cos ⁡ θ 2 ) 2\sqrt{\frac Rg}\cos^{-1}\left(\frac {1-\cos\frac\theta 2}{1+\cos\frac\theta 2}\right) 2 R g cos ⁡ − 1 ( 1 − sin ⁡ θ 2 1 + sin ⁡ θ 2 ) 2\sqrt{\frac Rg}\cos^{-1}\left(\frac {1-\sin\frac\theta 2}{1+\sin\frac\theta 2}\right)

Brian Lie by Brian Lie , 26, China

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1 solution

This not a solution, more an answer by inference. At θ = 0 , T = 0 \theta = 0, T = 0 . So, it can't be options B or C. At θ = 2 π \theta = 2\pi , option A gives bizarre results. Therefore, it's option D. :)

1 Helpful 1 Interesting 1 Brilliant 0 Confused

It should be at theta = 0 b is also ZERO And at theta equal to pi B holds good

raj abhinav - 1 year, 3 months ago

1 pending report

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