1 Suppose you were an intelligent small animal like an ant a

1. Suppose you were an intelligent, small animal (like an ant), and lived on the surface of a beach ball. Also, suppose that you only were aware of two dimensions (straight, back, left, and right), such that the concept of up and down did not exist. After class, you the small ant, decide to test whether you live on a curved surface, or whether your surface is flat, like the surface of a big pool table. Rather than walking in a straight line for a very long time and seeing if you end up where you started, you go out a buy a protractor. Then you draw three lines, making a big “triangle.” With the protractor, you measure the angle of “tip” of the triangle and add them together. If the angles add up to 180 degrees, you conclude your world is “flat,” and if not you conclude your world is “curved.” (a) Since you live on a beach ball, will the sum of the three angles be greater than or less than 180 degrees? (b) Discuss why you might come up with the wrong conclusion about the curvature of your world if you only make small triangles.

2. Suppose that a train robber decides to stop a train inside a tunnel. The proper length of the train is 60 m, while the proper length of the tunnel is 50 m. The train is traveling at 4/5 the speed of light. According to proper lengths, the train would not fit inside the tunnel. But the robber plans to use relativity to his advantage. The robber believes, due to length contraction, that he can trap the train inside the tunnel (by simultaneously blocking both ends of the tunnel). The engineer knows that his 60-m train will not fit completely into the tunnel, and does not worry about the robber. The robber thinks that the train will fit, whereas the engineer is sure it will not. Who is correct? Will the train fit inside the tunnel or not? Explain your reasoning.

Solution

2.

it will not fit because length contraction is from the point of view of an observer and does not reflect the actual position of the train.