Learning Goal:

To practice Problem$-$Solving Strategy $37\mathrm{.}1$ 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 $19\mathrm{.}0$ 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 $23\mathrm{.}0$ 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 cc $=3\mathrm{.}00108$ m$/$sm$/$s $.$ If they place their sample of the isotope on the spaceship, at what speed uu 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 $18\mathrm{.}0$ monthsmonths but it will still take $23\mathrm{.}0$ monthsmonths to build the new lab.

Problem Solving Strategy $37\mathrm{.}1$: Time Dilation

IDENTIFY the relevant concepts:

The concept of time dilation is used whenever you compare the time intervals between events as measured by observers in different inertial frames of reference.

SET UP the problem using the following steps:

To describe a time interval, you must first decide what two events define the beginning and the end of the interval. You must also identify the two frames of reference in which the time interval is measured.

Determine what the target variable is$.$

EXECUTE the solution as follows:

The proper time $$t$0$t$0$ is the time interval between two events in a frame of reference in which the two events occur at the same point in space. The dilated time $$t$$t is the longer time interval between the same two events as measured in a frame of reference that has a speed uu relative to the first frame. The two events occur at different points as measured in the second frame. You will need to decide in which frame the time interval is $$t$0$t$0$ and in which frame it is $$t$$t$.$

Use the equation

$$t$=$t$01$u$2/$c$2=$$$t$0$t$=$t$01$u$2/$c$2=$$$t$0$

to relate $$t$0$t$0$ and $$t$$t$,$ and then solve for the target variable.

EVALUATE your answer:

Note that $$t$$t is never smaller than $$t$0$t$0$ and uu is never greater than cc$.$ If your results suggest otherwise, you need to rethink your calculation.

IDENTIFY the relevant concepts

You must find the speed uu at which a spaceship needs to travel to allow a laboratory to be built by the time half of a sample of an isotope decays. Since the spaceship and the laboratory are in different reference frames, the above strategy is appropriate.

SET UP the problem using the following steps

Part A

Which two events define the proper time $$t$0$t$0?$

Check all that apply.

View Available Hint$($s$)$for Part A

The isotope leaves Earth on a spaceship traveling at speed uuThe new laboratory is completed.Construction begins on the new laboratoryHalf of the isotope stored on the spaceship decays.