5. For moving between points located hundreds of kilometers apart on the Earth's surface, people in the future will likely dig straight tunnels through which capsules will move without friction, solely under the influence of Earth's gravity. Let points $A, B$, and $C$ lie on the same meridian, and the distance from $A$ to $B$ along the surface is to the distance from $B$ to $C$ along the surface as $m: n$. A capsule travels through the tunnel $A B$ in approximately 42 minutes. Estimate the travel time through the tunnel $A C$. Provide your answer in minutes.
Answer: 42 min. Solution. Let point $O$ be the center of the Earth. To estimate the time of motion from $A$ to B, consider triangle $A O B$. We can assume that the angle $\alpha=90^{\circ}-\angle A B O$ is very small, so $\sin \alpha \approx \alpha$. Since the point in the tunnel $A B$ is attracted to the center by the gravitational force $F$, the projection of this force onto the direction of motion is $F \sin \alpha \approx F \alpha$. Under the action of this force, the body will move along the channel with acceleration $a \approx R \ddot{\alpha}$. Therefore, the model of the point's motion from $A$ to $B$ can be considered as the model of a mathematical pendulum, for which the period does not depend on the distance from $B$ to $A$. This means that the time of motion will be the same.

Answer to variant 162: 42 min.

Answer to variant 163: 42 min.

Answer to variant 164: 42 min.