One of the classic paradoxes of special relativity is the pole/barn paradox. The setup is: a person carrying a 20 m pole runs at relativistic speeds at a 15 m long barn (with doors open on both ends). If the person runs sufficiently fast enough, the pole will contract (according to the observer on the ground frame of reference, which we'll call S, like the textbook does) in length, according to the Lorentzian transformations, so that it will be short enough to fit inside the barn. To give a contrast, the door opening width in the barn (measured perpendicular to the runner's motion) is 5 m. However, from the person carrying the pole frame of reference (which we will call S'), it is the barn that contracts in length, with the pole staying 20 m long, and the pole will never fit inside the barn. Which point of view is correct? 1. Let the pole speed be 0.662c. Use the Lorentz transformation equations to calculate the following quantities (then fill in the table): S S' Frame of reference Length of pole Length of barn Width of barn door opening 2. Does the pole fit in the barn in frame S? 3. Draw a Minkowski diagram for the stationary frame of reference S: the axes are perpendicular to each other, with the x-axis being the x-coordinate and the y-axis being the ct-coordinate. Draw the walls of the barn as vertical lines, indicating that they are not moving, and set them 15 m apart. Mark the intersection of the x-axis with the left edge of the barn as origin. 4. Now draw the pole such that the front (right tip) of the pole is at origin when ct = 0. Draw the rest of the pole such that the length is what you calculated in question 1. 5. Draw the worldline of the pole by extending lines upward showing where the front and back tip of the pole will be at different times in the future. To do this, you will need to calculate the slope that corresponds to 0.662c. Hint: the 45° diagonal has a slope corresponding to c, and the worldlines of the pole should be steeper than that. Further hint: tan a =v/c, and you may need a protractor. 6. Draw a dark line that shows a point in time (in the stationary frame of reference) that when the pole is completely within the barn. This should coincide with the front tip of the pole reaching the front (right) edge of the barn. 7. Is the event of the front tip of the pole reaching the front edge of the barn simultaneous with the back tip of the pole entering the back edge of the barn, in the stationary frame of reference? 8. Now draw the ct' axis by emphasizing the line that already represents the worldline of the front (right) tip of the pole. The "*" notation indicates the moving frame of the pole. 9. Mirror that line across the 45° diagonal (the lightspeed line) to derive the x' axis. 10. Use a ruler and draw (using the scale you've developed) a 20 m line, representing the pole (from the pole frame of reference) whose front tip is coincident with the front tip of the pole you drew in question 5. Make sure that the line you draw is parallel to the x'axis! 11. Is the pole fully within the barn, from the pole frame of reference? In fact, is the event of the front tip of the pole reaching the front edge of the barn simultaneous with the back tip of the pole entering the back edge of the barn, from the pole frame of reference? 12. Conclusion: The problem is with the word Attach your Minkowski diagram to this sheet and turn in.