Summary Assignment for Chapter Four Name: ___Key____

This assignment is analogous to the calculations done in Chapter Four except that the light flash goes from the front of the bus to the back of the bus where Chuck is located. The bus moves with velocity v from left to right as usual. This experiment begins when Dean is adjacent to Bev and launches the light flash toward Chuck. As usual, the start of the experiment is defined to be at the point where X_bus = X_Earth = 0 and T_bus = T_Earth = 0. The experiment ends when the flash reaches Chuck who is adjacent to Anne at that instant.

The length of the bus according to Chuck and Dean is L_bus . Anne and Bev measure their separation to be D_Earth. Conversely, Chuck and Dean measure the separation between Anne and Bev to be D_bus while Anne and Bev measure the length of the bus to be L_Earth.

The Chuck and Dean Spacetime Diagram

1. At the start of the experiment, T_bus = 0, where is Chuck located? X_bus = _-L_bus_

2. At the start of the experiment, T_bus = 0, where is Dean located? X_bus = _0_

3. Draw the worldlines of Chuck and Dean on a spacetime diagram from their perspective. (Note that like in Chapter Four, this diagram does not have numbers but instead is defined in terms of distances defined above.)

4. Now draw the worldline of the laser flash. Label the place where the flash starts from as A and the point where it reaches Chuck as B.

5. Use points A and B to sketch in the worldlines of Anne and Bev.

6. At the start of the experiment, T_bus = 0, where is Anne located? X_bus = _-D_bus_

7. At the start of the experiment, T_bus = 0, where is Bev located? X_bus = _0_

Anne Chuck Bev Dean

-L_bus -D_bus 0

The Anne and Bev Spacetime Diagram

8. At the start of the experiment, T_Earth = 0, where is Anne located? X_Earth = __-D_Earth__

9. At the start of the experiment, T_Earth = 0, where is Bev located? X_Earth = _0_

10. Draw the worldlines of Anne and Bev on a spacetime diagram from their perspective.

11. Now draw the worldline of the laser flash. Label the place where the flash starts from as A and the point where it reaches Anne as B.

12. Use points A and B to sketch in the worldlines of chuck and Dean.

13. At the start of the experiment, T_Earth = 0, where is Chuck located? X_Earth = _-L_Earth_

14. At the start of the experiment, T_Earth = 0, where is Dean located? X_Earth = 0

Anne Chuck Bev Dean

-L_Earth -D_Earth 0

The goal is to use the two spacetime diagrams to find the shrinkage factor in terms of v. This endeavor uses the same strategy that Anne, Bev, Chuck, and Dean used in Chapter Four. So if you get a little confused, go back and review the arguments made by them in Chapter Four.

15. In terms of L_bus, L_Earth, D_bus, and D_Earth, what ratios are equal to the shrinkage factor?

Shrinkage Factor = L_Earth/L_bus = D_bus/D_Earth

16. Chuck and Dean use their spacetime diagram to derive an expression for D_bus in terms of L_bus, v, and the speed of light, c. What is that expression?

D_bus equals L_bus minus the distance Anne moves during the time it takes the light flash to travel from Dean to Chuck, L_bus/c.

D_bus = L_bus - v(L_bus/c)

17. Anne and Bev use their spacetime diagram to derive an expression for D_Earth in terms of L_Earth, v, and the speed of light, c. What is that expression?

D_Earth is L_Earth minus the distance Chuck moves during the time it takes the light flash to go from Bev to Anne, D_Earth/c.

D_Earth = L_Earth – v(D_Earth/c) which gives when solved for D_Earth, D_Earth = L_Earth/(1 + v/c).

18. Use your answers for questions 15, 16, and 17 to find the shrinkage factor.

L_Earth/L_bus = D_bus/D_Earth = [L_bus(1 – v/c)]/[L_Earth/(1 + v/c)] = (L_bus/L_Earth)[1 – (v/c)²]. Or finally, by moving the (L_bus/L_Earth) to the other side of the equation, (L_Earth/L_bus)² = [1 – (v/c)²]. Taking the square root gives the shrinkage factor, (L_Earth/L_bus) = .

The next part of the assignment is to use the spacetime diagrams to find the affect velocity has on the ticking rate of watches. At the start of the experiment, point A on the spacetime graphs, Bev is adjacent to Dean. Later in the experiment, Bev and Chuck are adjacent.

19. Find the point on the spacetime diagrams where Bev and Chuck are adjacent. Label that point C on both diagrams. You may have to modify the spacetime diagrams to include point C.

20. According to Chuck and Dean, what is the time on Chuck’s watch when Bev and Dean are adjacent?

Bev and Dean are adjacent at the start of the experiment so T_bus = 0.

21. According to Chuck and Dean, what is the time on Bev’s watch when Bev and Dean are adjacent?

Bev’s watch reads zero since it is the beginning of the experiment, T_Earth = 0

22. According to Chuck and Dean, how far does Bev have to travel to reach Chuck, point C on the spacetime diagram? _L_bus_

23. Use your answer to question 22 to find the time on Chuck’s watch when he is adjacent to Bev, point C.

Bev travels a distance L_bus with speed v, so T_bus = L_bus/v

24. According to Chuck and Dean, how long did it take Bev to travel from Dean to Chuck?

Since Dean’s watch read zero when Bev started the trip, the total time is just the reading on Chuck’s watch, L_bus/v.

Time according to Chuck and Dean = L_bus/v

25. When Bev is adjacent to Dean, Anne and Bev say that Chuck is located where on their spacetime diagram? X_Earth = _-L_Earth_

26. When Bev and Chuck are adjacent, point C, Bev’s watch reads what time?

Bev says that Chuck traveled the distance L_Earth with speed v, so T_Earth = L_Earth/v

27. How much time passes on Bev’s watch between her witnessing first Dean then Chuck passing? Bev’s watch read zero when Dean passed and L_Earth/v when Chuck passed, so T_Bev,Earth = L_Earth/v

28. On the other hand, according to watches worn by Chuck and Dean, it took Bev how long to move from Dean to Chuck? T_Bev,bus = L_bus/v

29. What is the ratio T_Bev,Earth/T_Bev,bus? (Your answer ought to be in terms of Lbus and LEarth.)

T_Bev,Earth/T_Bev,bus = L_Earth/L_bus

30. Use your answer to 18 to right the ratio of times in terms of v and c.

T_Bev,Earth/T_Bev,bus = .

31. In the above scenario, whose watch is the traveling watch that is being compared to two synchronized watches in a different reference frame?

Bev has the watch that is being compared to two watches in the bus frame, first Dean’s than Chuck’s.

32. Are your answers to questions 30 and 31 consistent with the idea that the “moving” watch, which measures proper time, ticks off less time than the two “stationary” watches?

Yes, because the “moving” watch is Bev’s and her watch ticks off less time and amount of less time is given by the shrinkage factor.