Question

Ann and Bob synchronize their wristwatches just as Ann departs the Earth in a rocketship headed toward the star system αCen at a speed v = 4 5 c. Bob remains on Earth. After one year, Bob sends a message (which travels at speed c) to Ann. Ann responds immediately. How much time has elapsed on Ann’s wristwatch when she receives Bob’s message? How long does Bob have to wait after sending the message to receive Ann’s response?

Answer 1

Distance of Cen from earth is 4.367 light years.

We know that 1 light year = 9.461 x 1015m.

So the distance of Cen from earth (D)= 4.367 x 9.461 x 1015m = 4.312 x 1016m.

Here Ann departs from earth at a velocity (vAnn) of 4.5c. So the time taken by Ann to reach Cen from earth is

That means Ann took 1.02yr to reach ​​​​​​​Cen from earth.

Now the message delivered after 1 yr has a velocity of light. If velocity of message is vm, then the time taken by the message to reach ​​​​​​​Cen from earth.

The message will take 4.6yr to reach ​​​​​​​Cen from earth. That means by the time the message reach Ann, she will be in Cen.

The message is sent 1 yr after Ann's departure. So the time elapsed on Ann's watch will be 5.6yr.

The message takes 4.6yr to reach Cen. So it takes twice that time for the response message to reach earth. So Bob has to wait 9.2 years to recieve the response message.