Question

At time t = 0, two starships set out from their homebase on Earth, which they all agree is at x = 0. Both ships head directly for planet LV-426, which is infested with vicious aliens. According to earth controllers, LV-426 is 3 lightyears away. The starship Flyby travels at a constant speed 0.8 c; its mission is to take photos of the situation on LV-426 as it flies past the planet. The second ship, the Sulacco, travels at a slower constant speed of 0.6 c. Its mission is to drop a bomb on the weapons factory of LV-426 as soon as it arrives. Two events are pivotal in this story: • Event A: the ships leave homebase • Event B: the Sulacco destroys the weapons factory on LV-426 Also, there are three observers: the Earth controllers, the Sulacco’s crew, and the Flyby’s crew.

(a) The story tells you the distance Δx and time Δt between events A and B in the Earth controllers’ reference frame. In discussion you calculated Δxʹ and Δtʹ in the Sulacco’s frame. Now, finish the job by calculating the distance Δxʹʹ and time Δtʹʹ between events A and B according to the Flyby’s crew. Collect all your values together and complete the following table: Earth frame S ulacco’s frame Flyby’s frame Δx Δt Interval

(b) Plot on the same graph at right the space-time coordinates of events A and B as seen by the three observers. To make this plot helpful, take event A as the common origin of all three observers’ coordinate systems: i.e. all three agree that event A occurred at x = 0 and t = 0. Roughly sketch the Lorentz-transformation hyperbola that connects the coordinates of event B as seen by all possible observers. Finally, indicate the interval on the graph: it is the distance between the origin (event A) and the hyperbola (event B) at a particular place … where?

Answer 1

a) Since we know the distance and time interval between Events A and B in the Earth frame, So we can use Lorentz transformations to calculate the same distance and time intervals in both the starships frames.

Frame	Distance(in light years)	Time (in years)
Earth	3	5
Sulacco	0	4
Flyby	1.667	4.333

The detailed calculations are done in the accompanied diagrams.
b) The space time diagram is attached in the image