Question

Suppose Anna and Carl travel to planet Y a distance of 27 LY away, and suppose they travel at 0.9c. Assume Bob remains at Earth. Anna approaches Planet Y from earth in one direction while Carl approaches Planet Y in the exact same distance away as Anna is, but approaches in the opposite direction. Answer the following questions (Explain fully to get full credit. Show any calculations):

(a) In Bob’s frame, Anna takes (27 LY)/0.9 = 30 years to get to Planet Y. In Anna’s frame, how many years does it take to get to Planet Y?

(b) As in the example in class, Carl and Anna pass each other at Planet Y each going 0.9c but in opposite directions. In Carl’s frame, how much has Bob aged since Carl passed Planet Y?

(c) Based on your answer in (b), what age would Carl conclude Bob was when he passed Planet Y?

(d) In Bob’s frame, how old is Bob when his 10th birthday message reaches Planet Y?

(e) In Carl’s frame, how long does it take Bob’s 10th birthday image to get from Earth to Planet Y? If Carl then calculates the time elapsed in Bob’s frame for the message to get to him, what will he conclude about how old Bob was when he was at Planet Y?

(f) In Anna’s frame, how long does it take the 10th birthday message to get from Earth to Planet Y?

(g) Since Anna knows about time dilation, she calculates how much time it took for the message to get from Earth to her at Planet Y. What would she conclude about Bob’s age when she was at Planet Y?

(h) In Carl’s frame, how long does it take for the 10th birthday message to get from Earth to Planet Y? What does he conclude about Bob’s age when he passed Planet Y?

Answer 1

a]

For Anna, the distance appears to be Lorentz contracted by a factor of 2.294 and she must cover this small distance with a speed of 0.9c.

so, the time taken as measured by Anna is:

t = 27/(2.294 x 0.9) = 13.076 years.

b] For Carl, the time it will take to reach Earth will be:

t = 27/(2.294 x 0.9) = 13.076 years.

so, in Carl's frame, Bob on earth would have aged by 13.076 years since Carl left planet Y.

c] If Bob's age was T when Carl reaches Earth, his age when Carl left planet Y would have been: T - 13.076 years.

d] The time it will take for Bob's message to reach planet Y will be:

t = 27 ly / c = 27 years.

[assuming the message was in the form of light pulses]

this message is sent on Bob's 10th birthday. So, his age when his 10th birthday message reaches planet Y will be:

T = 10+27 = 37 years.