Time Dilation. A group of scientists discover a new, rare isotope and are able to store a small amount of it. They determine that the isotope is unstable and half of their sample will decay in 13.0 monthsmonths . The scientists need a new laboratory to properly conduct measurements on the isotope. If laboratory is built before half the sample decays there will still be enough of the isotope available for experiments. However, it will take 21.0 monthsmonths to build the new lab. Fortunately, the scientists can quickly start construction and they have access to a spaceship that can travel at speeds approaching the speed of light c = 3.00

This question is incomplete, the complete question is;

Time Dilation. A group of scientists discover a new, rare isotope and are able to store a small amount of it. They determine that the isotope is unstable and half of their sample will decay in 13.0 months . The scientists need a new laboratory to properly conduct measurements on the isotope. If laboratory is built before half the sample decays there will still be enough of the isotope available for experiments. However, it will take 21.0 months to build the new lab. Fortunately, the scientists can quickly start construction and they have access to a spaceship that can travel at speeds approaching the speed of light c = 3.00 x 10⁸ m/s.

If they place their sample of the isotope on the spaceship, at what speed must it travel in order for the new laboratory to be completed on Earth by the time half of the isotope on the spaceship decays? Assume that it took one month for the scientists to actually start construction and launch the spaceship so half the sample will remain in 12.0 months but it will still take 21.0 months to build the new lab.

the required speed is 2.4618 x 10⁸ m/s

Explanation:

Given the data in the question;

Time taken to complete a lab = 21 months  

it took them one one to actually start so

Time remaining so that sample only decays by 1/2 = (13 - 1) = 12 months

we need to find at what speed the spaceship should travel so that time on earth ( = 12 months) become time on spaceship ( = 21 months)

we make use of time dilation equation;

t' = t/√( 1 - v²/c² )

where t is time in rest frame = 12 and t' is time in moving frame = 21

so we substitute

21 = 12/√( 1 - v²/c² )

21√( 1 - v²/c² ) = 12

√( 1 - v²/c² ) = 12/21

we square both side

( 1 - v²/c² ) = ( 12/21 )²

( 1 - v²/c² ) = 0.3265

v²/c² = 1 - 0.3265

v²/c² = 0.6735

v² = 0.6735 × c²

v = √0.6735 × √c²

v = 0.8206 × c

given that speed of light c = 3 x 10⁸ m/s

v = 0.8206 × 3 x 10⁸ m/s

v = 2.4618 x 10⁸ m/s

Therefore, the required speed is 2.4618 x 10⁸ m/s

+15-35=-20

going in opposite directions