1-20. Use the binomial expansion to derive the following results for values of $v \ll c$ and use when applicable in the problems that follow in this section. (a) $\gamma \approx 1 + \frac{1}{2}\frac{v^2}{c^2}$ (b) $\frac{1}{\gamma} \approx 1 - \frac{1}{2}\frac{v^2}{c^2}$ (c) $\gamma - 1 \approx \frac{1}{2}\frac{v^2}{c^2}$
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The Lorentz factor γ is defined as γ = 1/√(1-v²/c²), where v is velocity and c is the speed of light. Since we're told v << c, we can use binomial expansion to approximate this expression. Show more…
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Use the binomial expansion to derive the following results for values of $v \ll c$ and use when applicable in the problems that follow in this section. $$ \begin{array}{l} \text { (a) } \gamma \approx 1+\frac{1}{2} \frac{v^{2}}{c^{2}} \\ \text { (b) } \frac{1}{\gamma} \approx 1-\frac{1}{2} \frac{v^{2}}{c^{2}} \\ \text { (c) } \gamma-1 \approx 1-\frac{1}{\gamma} \approx \frac{1}{2} \frac{v^{2}}{c^{2}} \end{array} $$
(a) Using the binomial expansion (see Appendix I), show that Eq. $20-27$ reduces to the classical expression $K=\frac{1}{2} m v^{2}$ when $y<<c .(b)$ By evaluating the second term in the expansion, find the value of $v / c$ for which the error in using the classical expression is at most $1 \%$.
Let$$ X=\left({ }^{10} C_{1}\right)^{2}+2\left({ }^{10} C_{2}\right)^{2}+3\left({ }^{10} C_{3}\right)^{2}+\cdots+10\left({ }^{10} C_{10}\right)^{2}, $$ where ${ }^{10} C_{r} r \in\{1,2, \cdots, 10\}$ denote binomial coefficients. Then, the value of $\frac{1}{1430} X$ is
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