91Ó°ÊÓ

The length contraction equation states that ${\mathbf{Δ³\ae }}^{\mathbf{\text{'}}}\mathbf{=}\mathbf{γ}\mathbf{(}\mathbf{Δ³\ae }\mathbf{−}\mathbf{vΔ³Ù}\mathbf{)}$, where the expression$\mathbf{γ}$ is the Lorentz factor.

From the above formula, we can say that

$\begin{array}{c}{\mathrm{Δ³\ae }}^{\text{'}}=\mathrm{γ}(\mathrm{Δ³\ae }−\mathrm{vΔ³Ù})\\ =\mathrm{γ}(\mathrm{Δ³\ae }−\mathrm{βcΔ³Ù})\\ =\frac{400−300\mathrm{β}}{\sqrt{1−{\mathrm{β}}^2}}\end{array}$

Thus, the expression for${\mathrm{Δ³\ae }}^{\text{'}}$ is ${\mathrm{Δ³\ae }}^{\text{'}}=\frac{400−300\mathrm{β}}{\sqrt{1−{\mathrm{β}}^2}}$.

(b)

The plot of ${\mathrm{Δ³\ae }}^{\text{'}}$as a function of $\mathrm{β}$for $0<\mathrm{β}<0.01$is shown below:

(c)

The plot of${\mathrm{Δ³\ae }}^{\text{'}}$ as a function of$\mathrm{β}$ for$0.1<\mathrm{β}<1$ is shown below:

To calculate the value of$\mathrm{β}$where the value of$∆\mathrm{x}\text{'}$is minimum, we have to find the derivative of$∆\mathrm{x}\text{'}$and then put it equal to zero to solve for$\mathrm{β}$.

$\begin{array}{c}\frac{\mathrm{d}\left({\mathrm{Δ³\ae }}^{\text{'}}\right)}{\mathrm{»åβ}}=\frac{\mathrm{d}}{\mathrm{»åβ}}\left(\frac{\mathrm{Δ³\ae }−\mathrm{βcΔ³Ù}}{\sqrt{1−{\mathrm{β}}^2}}\right)\\ =\frac{\mathrm{βΔ³\ae }−\mathrm{cΔ³Ù}}{{(1−{\mathrm{β}}^2)}^{\frac{3}{2}}}\end{array}$

Now, put it equal to zero:

$\begin{array}{c}\frac{\mathrm{βΔ³\ae }−\mathrm{cΔ³Ù}}{{(1−{\mathrm{β}}^2)}^{\frac{3}{2}}}=0\\ \mathrm{βΔ³\ae }−\mathrm{cΔ³Ù}=0\\ \mathrm{β}=\frac{\mathrm{cΔ³Ù}}{\mathrm{Δ³¦³\ae }}\\ \mathrm{β}=\frac{(3.00×{10}^8)(1.0×{10}^{−6})}{400}\\ \mathrm{β}=0.750\end{array}$

Thus, the minimum value of $∆\mathrm{x}\text{'}$lies at $\mathrm{β}=0.750$.

(e)

At $\mathrm{β}=0.750$, the value of $∆\mathrm{x}\text{'}$is:

$\begin{array}{c}{\mathrm{Δ³\ae }}^{\text{'}}=\frac{400−\left(300\right)\mathrm{β}}{\sqrt{1−{\mathrm{β}}^2}}\\ =\frac{400−300\left(0.75\right)}{\sqrt{1−{\left(0.75\right)}^2}}\\ =264.8\text{″¾}\end{array}$

Thus, the minimum value of$∆\mathrm{x}\text{'}$ is approximately$265\text{″¾}$ .