As a research scientist at a linear accelerator, you are studying an unstable particle. You measure its mean lifetime$\mathbf{∆}\mathit{t}$ as a function of the particle’s speed relative to your laboratory equipment. You record the speed of the particleu as a fraction of the speed of light in vacuum c. The table gives the results of your measurements.

Your team leader suggests that if you plot your data as${\left(∆t\right)}^{\mathbf{2}}$versus${\left(1-\frac{u^2}{c^2}\right)}^{\mathbf{-}\mathbf{1}}$, the data points will be fit well by a straight line. Construct this graph and verify the team leader’s prediction. Use the best-fit straight line to your data to calculate the mean lifetime of the particle in its rest frame. (b) What is the speed of the particle relative to your lab equipment (expressed as$\frac{\mathbf{u}}{\mathbf{c}}$) if the lifetime that you measure is four times its rest-frame lifetime?

Step by step solution

01

Construct the graph

The following figure shows the best-fit straight line to the data. We see$R^2=0.998$ (the correlation coefficient R is very close to unity), which means the plotted data is well fit by a straight line. The corresponding equation of the fit straight line is$(\mathrm{\Delta }t{)}^2=\left(6.78\times {10}^{-16}\right){\left(1-\frac{u^2}{c^2}\right)}^{-1}$

The slope of this line is equal to square of the rest-frame lifetime $∆t_0$, so we find$∆t_0=\sqrt{6.78\times {10}^{-16}}=2.6\times {10}^{-8}s$

02

Calculate the relative speed of the particle

Let us first express the computed equation of the fit line in terms of $∆t_0$:

$(\mathrm{\Delta }t{)}^2={\left(\mathrm{\Delta }{t}_0\right)}^2{\left(1-\frac{u^2}{c^2}\right)}^{-1}$

Solving this equation for $\frac{u}{c}$gives $\frac{u}{c}=\sqrt{1-{\left(\frac{\mathrm{\Delta }{t}_0}{\mathrm{\Delta }t}\right)}^2}$

For $\Delta t=4\Delta t_0$, we find $\frac{u}{c}=0.97.$

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