# Revolutionary Riddles: Answers

3) Two laps around a track.

There are two interpretations for this one. First is speed, the other is velocity.

Speed is a scalar quanitity; it is the magnitude of the instantaneous velocity vector. If you integrate speed with respect to time, you get the total distance traveled, regardless of path. Average speed is total distance traveled, divided by total time.

So in the speed interpretation the answer is that there is no possible way to have an average speed of twice your first lap speed. On your first lap, your speed L/t_1. Your average speed is 2L/t_total. Since the total distance is twice the single lap distance, for the average speed to be double the single lap speed, the total time would have to be the same for both the first lap and both laps combined, so there is no finite speed at which you can travel to achieve this.

Velocity is a vector quantity. If you integrate the instantaneous velocity with respect to time, you get the final offset from the initial starting position. This vector is called displacement. The average velocity is the displacement divided by the total time. Note that the magnitude of the average velocity is not equal to the average speed.

In the velocity interpretation, since the track forms a closed loop, the start and end positions are the same. Therefore, the total displacement is zero. This means that the velocity of each lap is always zero, so your average velocity is trivially equal to twice your single lap velocity, no matter how fast you run your second lap.

4) Part of a train is moving backwards.

Train wheels have flanges on them, which extend below the contact surface of the wheel on the track. In a normal/ideal wheel, the bottom of the wheel has a speed of zero relative to the ground and the top has a speed of twice the axle speed. Since the flange extends a little beyond the contact point, it will move slightly backwards beneath the contact point.

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