Question

piloting a state-of-the-art spaceship to α Centauri, which is a star about 4 lightyears away as measured from Earth. Your ship cruises at a speed of 0.6c. For this problem, you can assume that Earth and a Centauri are at rest with respect to each other. At your journey's midpoint, you decide to send a souvenir package to your folks back home. You launcha probe back to Earth at a speed of v 0.8c from your perspective. a. How long does the package take to arrive as seen from your perspective? b. How fast is the package going in Earth's frame of reference? c. Use your answer to part b) to find out how long it takes for the package to arrive in Earth's frame. After a year has passed in your perspective, the package gets torn apart by stray interstellar debris. About half the gift is left intact and moving towards Earth. From Earth's frame, the leftover half of the gift is moving at v0.87c towards Earth. d. How fast is the rest of the gift package moving from the gift's original frame (i.e. before it got destroyed)? e. How long will this remnant take to reach Earth from our viewpoint?

Answer 1

a) The mid point is 2 light years away from Earth's perspective. From my perspective it is

2V1-0.62-1.6

L = 1.6 light years

the speed of probe is 0.8 c. Therefore it will take 1.6/0.8 = 2 years

Answer: 2 years

b) The relative speed as seen by me is given by

$u' = \frac{u-v}{1-\frac{uv}{c^2}}$

0.8c _u0.6c

uー-cー0.384620 13

Answer: 0.38462 c

c) In Earth's frame the package will take

years

= 5.2 13

Answer: 5.2 years

d) By law of conservation of linear momentum the other pack is also moving at 0.87c in the other direction.

Therefore relative velocity between them

0.87c-(-0.87c)

Answer: 0.9904 c

e) From my viewpoint the package is travelling at speed

0.6c)-0.9660 0.87c(-0.6c)一一0.966C 1

In my frame when I reach half the distance Earth is

light years away.

2V1-0.62-1.6

For 1st 1 year the package is moving at 0.8 c therefore when the package breaks it is

1.6-0.8*1 = 0.8 light years away

at 0.966 c the package will take

0.8/0.966 = 0.8283 years

Answer: 0.8283 years