Chapter 26, Problem 3P

(a)

To determine

The factor for the given speed.

Answer to Problem 3P

  $\text{1}\text{.00}$ .

Explanation of Solution

Given:

Speed, $v\text{=}\text{20,00}0\text{m/s}$

Formula used:

The relativistic factor has the given form: $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$ or ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

Where,

v is velocity

c is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

Substituting the given values in the formula: ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

  $\begin{array}{c}f\text{=}{\left\{\text{1}-{\left[\frac{\text{20,000}\text{m/s}}{\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}}\right]}^{\text{2}}\right\}}^{\frac{\text{1}}{\text{2}}}\\ \text{=1}\text{.00}\end{array}$

(b)

To determine

The factor for the given speed.

Answer to Problem 3P

  $\text{0}\text{.9998}$ .

Explanation of Solution

Given:

Speed, $v=0.020c$

Formula used:

The relativistic factor has the given form: $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$ or ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

Where,

v is velocity

c is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

Substituting the given values in the formula: ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

  $\begin{array}{c}f\text{=}{\left\{\text{1-}{\left[\frac{\text{0}\text{.020}\text{c}}{\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}}\right]}^{\text{2}}\right\}}^{\frac{\text{1}}{\text{2}}}\\ \text{=0}\text{.9998}\end{array}$

(c)

To determine

The factor for the given speed.

Answer to Problem 3P

  $\text{0}\text{.980}$ .

Explanation of Solution

Given:

Speed, $v=0.200c$

Formula used:

The relativistic factor has the given form: $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$ or ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

Where,

v is velocity

c is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

Substituting the given values in the formula: ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

  $\begin{array}{c}f\text{=}{\left\{\text{1-}{\left[\frac{\text{0}\text{.200}\text{c}}{\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}}\right]}^{\text{2}}\right\}}^{\frac{\text{1}}{\text{2}}}\\ \text{=0}\text{.980}\end{array}$

(d)

To determine

The factor for the given speed.

Answer to Problem 3P

  $\text{0}\text{.312}$ .

Explanation of Solution

Given:

Speed, $v=0.95c$

Formula used:

The relativistic factor has the given form: $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$ or ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

Where,

v is velocity

c is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

Substituting the given values in the formula: ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

  $\begin{array}{c}f\text{=}{\left\{\text{1-}{\left[\frac{\text{0}\text{.95}\text{c}}{\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}}\right]}^{\text{2}}\right\}}^{\frac{\text{1}}{\text{2}}}\\ \text{=0}\text{.312}\end{array}$

(e)

To determine

The factor for the given speed.

Answer to Problem 3P

  $\text{0}\text{.199}$ .

Explanation of Solution

Given:

Speed, $v=0.98c$

Formula used:

The relativistic factor has the given form: $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$ or ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

Where,

v is velocity

c is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

Substituting the given values in the formula: ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

  $\begin{array}{c}f\text{=}{\left\{\text{1-}{\left[\frac{\text{0}\text{.98}\text{c}}{\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}}\right]}^{\text{2}}\right\}}^{\frac{\text{1}}{\text{2}}}\\ \text{=0}\text{.199}\end{array}$

(f)

To determine

The factor for the given speed.

Answer to Problem 3P

  $\text{0}\text{.0447}$ .

Explanation of Solution

Given:

Speed, $v=0.999c$

Formula used:

The relativistic factor has the given form: $\sqrt{1-{\left(\frac{v}{c}\right)}^2}$ or ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

Where,

v is velocity

c is speed of light, $3.00\times {10}^{8}m/s$

Calculation:

Substituting the given values in the formula: ${\left[1-{\left(\frac{v}{c}\right)}^2\right]}^{\frac{1}{2}}$

  $\begin{array}{c}f\text{=}{\left\{\text{1-}{\left[\frac{\text{0}\text{.999}\text{c}}{\text{3}{\text{.00×10}}^{\text{8}}\text{m/s}}\right]}^{\text{2}}\right\}}^{\frac{\text{1}}{\text{2}}}\\ \text{=0}\text{.0447}\end{array}$