- #1
lowea001
- 29
- 3
Homework Statement
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Two cities are connected by a straight underground tunnel, as shown in the diagram. A train starting from rest travels between the two cities powered only by the gravitational force of the Earth, [itex]F = - \frac{mgr}{R}[/itex].
Find the time [itex]t_1[/itex] taken to travel between the two cities (i.e. half the period). The distance between the two cities is d and the radius of the Earth is R. Now, suppose the train is given an initial an initial velocity [itex]v_0[/itex]. What is [itex]v_0[/itex] if the time taken to reach the other end of the tunnel is now [itex]t_2 = \frac{t_1}{2}[/itex].
Hint: Since [itex]t_1 = \frac{T_1}{2} [/itex] that means [itex]t_1 = \frac{1}{2} \frac{2 \pi}{w} = \frac{\pi}{w}[/itex] and therefore [itex]t_2 = \frac{\pi}{2w}[/itex].
I have the answer to this question I just don't know how to do it.
Homework Equations
Simple Harmonic Motion, Separable DE, Second-order DE, Newton's Second Law, Chain Rule
The Attempt at a Solution
I can get an equation for [itex]v_0[/itex] in terms of x and v but I don't know how to get that in terms of t or if I'm even approaching this the right way.