Question

A vessel is launched from Earth, carrying a population of goats at relativistic speed toward a colony on a far away planet. The goats won't breed during spaceflight, and they have an expected lifetime of 15.0 years before they start to die of old age. How far in light years (ly) can the vessel travel, in the reference frame of Earth, before the population of goats begins to die off... if the vessel travels at 0.8450c? if the vessel travels at 0.9800c?

Answer 1

Since the can is travelling with the comparable speed of light(c) thus in this we need to use the concept of relativity. Then according to that

Time dilation or time taken by goats to die of old age will be $t'= \frac{t}{\sqrt{1-\frac{v^2}{c^2}}}$

a). Then if the speed is v= 0.845c then

Time taken, $t'= \frac{15}{\sqrt{1-\frac{(0.845c)^2}{c^2}}}$

   $t'= \frac{15}{\sqrt{1-0.714}}= \frac{15}{0.534}= 28.08 yrs$

Then distance travelled in this much time is

$D'= vt'= 0.854 \times 3 \times 10^8 \times 28.08 \times 365 \times 24 \times 60 \times 60$

$D'= 2.268 \times 10^{17} m= (2.268 \times 10^{17} \times 1.057 \times 10^{-16}) ly$

(ANS)

b). Now if the speed is v= 0.98c then

Time taken, $t'= \frac{15}{\sqrt{1-\frac{(0.98c)^2}{c^2}}}$

   $t'= \frac{15}{\sqrt{1-0.96}}= \frac{15}{0.2}= 75 yrs$

Then distance travelled in this much time is

$D'= vt'= 0.98 \times 3 \times 10^8 \times 75 \times 365 \times 24 \times 60 \times 60$

$D'= 6.953 \times 10^{17} m= (6.953 \times 10^{17} \times 1.057 \times 10^{-16}) ly$

(ANS)