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Chapter 33: Problem 34

Radioactive oxygen-15 decays at such a rate that half the atoms in a given sample decay every 2 min. If a tube containing 1000 O-15 atoms is moved at \(0.80 c\) relative to Earth for 6.67 min according to clocks on Earth, how many atoms will be left at the end of that time?

Short Answer

Expert verified
After considering both, the radioactive decay and the relativistic time dilation, approximately 177 Oxygen-15 atoms will be left at the end of the experiment as observed from Earth.

Step by step solution

01

Calculate Decay Time in the Moving Frame

Initially, identify the decay half-life given, which is 2 minutes. The problem states that the observer on Earth measures the time of the experiment as 6.67 minutes. Calculating how many half lives this corresponds to will provide the decay time in the moving frame. Divide the total time measured on Earth by the half-life, resulting in \(6.67 min ÷ 2 min = 3.335\) half-life periods.
02

Apply the Formula of Radioactive Decay

Radioactive decay follows an exponential process determined by the formula: \(N = N_0 (1/2)^n\), where \(N\) is the final amount of substance, \(N_0\) is the initial amount, and \(n\) is the number of half-life periods. Here \(N_0 = 1000\) atoms and \(n = 3.335\). Plug these values into the decay formula to calculate \(N\) atoms.
03

Consider Time Dilation Due to Theory of Relativity

Because the sample is moving at a speed close to the speed of light, time dilation must be considered. According to the Theory of Relativity, time would progress slower for the moving sample from the perspective of the observer on Earth. The time dilation factor is given by \(\sqrt{1 - (v^2/c^2)}\), where \(v = 0.80c\). Substitute this into the time dilation formula to find the dilated time, and then apply the decay formula as in step 2.

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