Question

What is the speed parameter for the following speeds: (a) a typical rate of continental drift (1 in./y); (b) a typical drift speed for electrons in a current-carrying conductor (0.5 mm/s); (c) a highway speed limit of 55 mi/h; (d) the root-mean-square speed of a hydrogen molecule at room temperature; (e) a supersonic plane flying at Mach 2.5 (1200 km/h); (f) the escape speed of a projectile from the Earth’s surface; (g) the speed of Earth in its orbit around the Sun; (h) a typical recession of a distant quasar due to the cosmological expansion $\mathbf{3}\mathbf{\times }{\mathbf{10}}^{\mathbf{4}}\mathbf{}\mathit{k}\mathit{m}{\mathit{s}}^{\mathbf{-}\mathbf{1}}$.

Short answer

(a) The speed parameter of continental drift is $3\times {10}^{-18}$.

(b) The speed parameter of drift speed of electron is $2\times {10}^{-12}$.

(c) The speed parameter for highway speed limit is $8.2\times {10}^{-8}$.

(d) The speed parameter for root mean square speed of hydrogen molecule is $8\times {10}^{-6}$.

(e) The speed parameter for supersonic flying plane is $1.1\times {10}^{-6}$.

(f) The speed parameter for projectile from Earth’s surface is $3.7\times {10}^{-5}$

(g) The speed parameter for earth in its orbit around sun is $9.9\times {10}^{-5}$.

(h) The speed parameter for distant quasar is 0.10.

Step-by-step solution

Identification of given data

The rate of continental drift is $V_c=1\mathrm{m}/y$

The drift speed of electron isrole="math" localid="1663066400474" $V_d=05\mathrm{mm}ls$

The highway speed limit is $V_h=55mi/h$

The speed of supersonic plane is role="math" localid="1663066459491" ${\mathrm{V}}_{\mathrm{s}}=110\mathrm{km}lh$

The rate of cosmological expansion is$u=3\times {10}^{4}km/s$

The speed parameter is the ratio of speed of particle with speed of light. It describes the comparison of the particle with light particles.

Determination of speed parameter for continental drift

(a)

The speed parameter of continental drift is given as:

$\beta =\frac{V_c}{c}$

Here, c is the speed of light and its value is $3\times {10}^{8}m/s$.

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(1\text{in}/\text{y}\right)\left(\frac{0.0254\text{m}}{1\text{in}}\right)\left(\frac{1\text{y}}{3.154\times {10}^7\text{s}}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =3\times {10}^{-18} \end{gathered}$$

Therefore, the speed parameter of continental drift is $3\times {10}^{-18}$.

Determination of speed parameter for drift speed of electron

(b)

The speed parameter of drift speed of electron is given as:

$\beta =\frac{V_d}{c}$

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(0.5\text{mm}/\text{s}\right)\left(\frac{1\text{m}}{1000\text{mm}}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =2\times {10}^{-12} \end{gathered}$$

Therefore, the speed parameter of drift speed of electron is $2\times {10}^{-12}$.

Determination of speed parameter for highway speed limit

(c)

The speed parameter for highway speed limit is given as:

$\beta =\frac{V_h}{c}$

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(55\text{mi}/\text{h}\right)\left(\frac{1600\text{m}}{1\text{mi}}\right)\left(\frac{1\text{h}}{3600\text{s}}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =8.2\times {10}^{-8} \end{gathered}$$

Therefore, the speed parameter for highway speed limit is $8.2\times {10}^{-8}$.

Determination of speed parameter for root mean square speed of hydrogen molecule

(d)

The speed parameter for root mean square speed of hydrogen molecule is given as:

$\beta =\frac{V_r}{c}$

Here, $v_r$is the root mean square speed of hydrogen molecule at room temperature and its value is $2400m/s$.

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(2400\text{m}/\text{s}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =8\times {10}^{-6} \end{gathered}$$

Therefore, the speed parameter for root mean square speed of hydrogen molecule is $8\times {10}^{-6}$.

Determination of speed parameter for supersonic flying plane

The speed parameter for supersonic flying plane is given as:

$\beta =\frac{V_s}{c}$

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(1200\text{km}/\text{h}\right)\left(\frac{1000\text{m}}{1\text{km}}\right)\left(\frac{1\text{h}}{3600\text{s}}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =1.1\times {10}^{-6} \end{gathered}$$

Therefore, the speed parameter for supersonic flying plane is $1.1\times {10}^{-6}$.

Determination of speed parameter for projectile from Earth’s surface

(f)

The speed parameter for projectile from Earth’s surface is given as:

$\beta =\frac{V_e}{c}$

Here, $v_e$is the escape speed of projectile on Earth and its value is $11200m/s$.

Substitute all the values in the above equation.

role="math" localid="1663067724544" $$\begin{gathered} \beta =\frac{\left(11200\text{m}/\text{s}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =3.7\times {10}^{-5} \end{gathered}$$

Therefore, the speed parameter for projectile from Earth’s surface is$3.7\times {10}^{-5}$

Determination of speed parameter for Earth in its orbit around sun

(g)

The speed parameter for earth in its orbit around sun is given as:

$\beta =\frac{V_o}{c}$

Here, $v_0$is the orbital speed of Earth around sun and its value is $297000m/s$.

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(297000\text{m}/\text{s}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =9.9\times {10}^{-5} \end{gathered}$$

Therefore, the speed parameter for earth in its orbit around sun is $9.9\times {10}^{-5}$

Determination of speed parameter for supersonic flying plane

(h)

The speed parameter for distant quasar is given as:

$\beta =\frac{u}{c}$

Substitute all the values in the above equation.

$$\begin{gathered} \beta =\frac{\left(3\times {10}^4\text{km}/\text{s}\right)\left(\frac{{10}^3\text{m}}{1\text{km}}\right)}{c\left(\frac{3\times {10}^8\text{m}/\text{s}}{c}\right)} \\ =0.10 \end{gathered}$$

Therefore, the speed parameter for distant quasar is 0.10.