Chapter 26, Problem 10P

(a)

To determine

Speed at which the relativistic formula for length differs from classical value by the given percentage.

Answer to Problem 10P

Speed, $v=0.141c$

Explanation of Solution

Given:

Relativistic formula for length differs from classical value by 1.00%.

Formula used:

Length contraction can be given by:

  $l=l_0\sqrt{1-\frac{v^2}{c^2}}$

Calculation:

Rearranging the above formula,

  $v=c\sqrt{1-{\left(\frac{l}{l_0}\right)}^2}$

Now, according to the given data,

  $\begin{array}{l}\frac{l}{l_0}=100\%-1.00\%=99\%\\ \frac{l}{l_0}=0.99\end{array}$

Putting the value in the expression,

  $\begin{array}{l}v=c\sqrt{1-{\left(0.99\right)}^2}\\ v=0.141c\end{array}$

Conclusion:

Hence, the speed at which the relativistic formula for length differs from classical value by 1.00% is $v=0.141c$

(b)

To determine

Speed at which the relativistic formula for time interval differs from classical value by given percentage.

Answer to Problem 10P

Speed, $v=0.140c$

Explanation of Solution

Given:

Relativistic formula for time interval differs from classical value by 1.00%.

Formula used:

Time dilation can be given by:

  $\Delta t_0=\Delta t\sqrt{1-\frac{v^2}{c^2}}$

Calculation:

Rearranging the above formula,

  $v=c\sqrt{1-{\left(\frac{\Delta t_0}{\Delta t}\right)}^2}$

According to the given data,

  $\begin{array}{l}\frac{\Delta t}{\Delta t_0}=\left(100\%+1\%\right)=101\%\\ \frac{\Delta t}{\Delta t_0}=1.01\\ \frac{\Delta t_0}{\Delta t}=\frac{1}{1.01}\end{array}$

Now, substituting the value in the formula,

  $\begin{array}{l}v=c\sqrt{1-{\left(\frac{1}{1.01}\right)}^2}\\ v=0.140c\end{array}$

Conclusion:

Thus, the speed at which the relativistic formula for time interval differs from classical value by 1.00% is $v=0.140c$