Question

this would require digging a tunnel through the earth from boston to frankfurt, which has a surface distance of 5900 km. the train would move down the first part of the tunnel under the pull of gravity and then coast upward against gravity. assuming the motion will be simple harmonic motion, how long will a one-way trip take?

Answer 1

Performing this calculation requires making some simplifying assumptions:

Assumptions:

1. The tunnel is frictionless so that the train can move simply based on the gravitational force. In reality there would be friction that would slow the train down.

2. The tunnel is a straight line through the center of the Earth. In reality it would likely take a curved path.

3. The motion of the train follows simple harmonic motion. This is an approximation that ignores non-linear effects.

Given these assumptions, we can calculate the time for a one-way trip as follows:

- The radius of the Earth is 6371 km

- The distance the train needs to travel is 5900 km, or 92.6% of the Earth's radius

- Using simple harmonic motion, the time for 1 oscillation is T = 2π(R/g)^0.5 where R is the maximum distance and g is acceleration due to gravity (9.8 m/s^2)

- Plugging in the values, the time for 1 full oscillation is T = 42 minutes

- Since the train only needs to travel 92.6% of the radius, it will take 92.6% of a full oscillation, or 39 minutes.

So in summary, under the assumptions made, the one-way trip time from Boston to Frankfurt through the Earth's core would be approximately 39 minutes. In reality, friction and non-linear effects would slow the train down and make the trip take significantly longer.