Question

A car is traveling along the line in S (Fig. 12.25), at (ordinary) speed$\left(2/\sqrt{5}\right)$c .

(a) Find the components U_x and U_yof the (ordinary) velocity.

(b) Find the components ${\eta }_x$and${\eta }_y$of the proper velocity.

(c) Find the zeroth component of the 4-velocity, ${\eta }^0$.

System $\overline{S}$is moving in the x direction with (ordinary) speed ,$\sqrt{2/5}c$ relative to S. By using the appropriate transformation laws:

(d) Find the (ordinary) velocity components ${\upsilon }_x$and${\upsilon }_y$in $\overline{S}$.

(e) Find the proper velocity components ${\eta }_{x}and{\eta }_y$in $\overline{S}$.

(f) As a consistency check, verify that

$\overline{\eta }=\frac{\overline{u}}{\sqrt{1-{\displaystyle \frac{{\overline{u}}^2}{c^2}}}}$

Short answer

(a) The components${\upsilon }_x$and${\upsilon }_y$of the ordinary velocity are$\sqrt{\frac{2}{5}}c$.

(b) The components${\eta }_x$and${\eta }_y$of the proper velocity are $\sqrt{2}c$.

(c) The zeroth component of the 4-velocity is ${\eta }^0=\sqrt{5}c$.

(d) The components $\overline{u_x}$and$\overline{u_y}$of the ordinary velocity in $\overline{S}$are 0 and$\sqrt{\frac{2}{3}}c$respectively.

(e) The components $\overline{{\eta }_x}$and $\overline{{\eta }_y}$of the proper velocity in$\overline{S}$are 0 and $\sqrt{2}c$respectively.

(f) It is proved that $\overline{\eta }=\frac{\overline{u}}{\sqrt{1-{\displaystyle \frac{{\overline{u}}^2}{c^2}}}}$.

Step-by-step solution

Given information

Given data:

The ordinary speed is $u=\frac{2}{\sqrt{5}}c$.

The angle of the line in S is$\theta =45°$ .

Determine the components υx and υy of the velocity: 

(a)

Write the x-component of the ordinary velocity.

${\upsilon }_x=u\cos \theta$

Substitute $\frac{2}{\sqrt{5}}c$ for u and 45^o for $\theta$in the above equation.

$$\begin{gathered} {\upsilon }_x=\left(\frac{2}{\sqrt{5}}c\right)\cos \left(45°\right) \\ {\upsilon }_x=\left(\frac{2}{\sqrt{5}}c\right)\left(\frac{1}{\sqrt{2}}\right)\times \left(\frac{\sqrt{2}}{\sqrt{2}}\right) \\ {\upsilon }_x=\frac{2\sqrt{2}}{2\sqrt{5}} \\ {\upsilon }_x=\sqrt{\frac{2}{5}}c \end{gathered}$$

Write the y-component of the ordinary velocity.

${\upsilon }_y=\upsilon \cos \theta$

Substitute $\frac{2}{\sqrt{5}}c$for $u$ and 45^o for $\theta$in the above equation.

$$\begin{gathered} {\upsilon }_y=\left(\frac{2}{\sqrt{5}}c\right)\sin \left(45°\right) \\ {\upsilon }_y=\left(\frac{2}{\sqrt{5}}c\right)\left(\frac{1}{\sqrt{2}}\right)\times \left(\frac{\sqrt{2}}{\sqrt{2}}\right) \\ {\upsilon }_y=\frac{2\sqrt{2}}{2\sqrt{5}}c \\ {\upsilon }_y=\sqrt{\frac{2}{5}}c \end{gathered}$$

Therefore, the components ${\upsilon }_x$and ${\upsilon }_y$of the ordinary velocity are $\sqrt{\frac{2}{5}}c$.

Determine the componentsηx and ηy and of the proper velocity: 

(b)

Write the expression for the proper velocity.

$\eta =\frac{u}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$

Write the x-component of the proper velocity.

${\eta }_x=\frac{{\upsilon }_x}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$

Substitute $\sqrt{\frac{2}{5}}c$for $u_x$and $\frac{2}{\sqrt{5}}c$for $u$ in the above equation.

$$\begin{gathered} {\eta }_x=\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{\sqrt{1-{\displaystyle \frac{{\left({\displaystyle \frac{2}{\sqrt{5}}}c\right)}^2}{c^2}}}} \\ =\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{\sqrt{1-{\displaystyle \frac{4}{5}}}} \\ =\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{\sqrt{{\displaystyle \frac{1}{5}}}} \\ =\sqrt{2}c \end{gathered}$$

Write the y-component of the proper velocity.

${\eta }_y=\frac{{\upsilon }_y}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$

Substitute $\sqrt{\frac{2}{5}}c$for $u_y$and $\frac{2}{\sqrt{5}}c$for u in the above equation.

$$\begin{gathered} {\eta }_y=\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{\sqrt{1-{\displaystyle \frac{{\left({\displaystyle \frac{2}{\sqrt{5}}}c\right)}^2}{c^2}}}} \\ {n}_y=\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{\sqrt{1-{\displaystyle \frac{4}{5}}}} \\ {n}_y=\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{\sqrt{{\displaystyle \frac{1}{5}}}}=\sqrt{2}c \end{gathered}$$

Therefore, the components of the proper velocity are$\sqrt{2}c$ .

Determine the zeroth component of the 4-velocity, ηo

(c)

Write the expression for the zeroth components of the 4-velocity.

${\eta }^o=\frac{c}{\sqrt{1-{\displaystyle \frac{u^2}{c^2}}}}$ …… (1)

It is known that:

$u=\sqrt{{{\eta }_x}^2+{{\eta }_y}^2}$ …… (2)

Substitute $\sqrt{2}c$for ${\eta }_x$and $\eta$in equation (2).

$$\begin{gathered} u=\sqrt{{\left(\sqrt{2c}\right)}^2+{\left(\sqrt{2c}\right)}^2} \\ =\sqrt{2c^2+2c^2} \\ =\sqrt{4}c \\ =2c \end{gathered}$$

Substitute $\sqrt{\frac{2}{5}}c$for $\upsilon$in equation (1).

$$\begin{gathered} \eta °=\frac{c}{\sqrt{1-{\displaystyle \frac{{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}^2}{c^2}}}} \\ =\frac{c}{\sqrt{1-{\displaystyle \frac{4}{5}}}} \\ =\frac{c}{\sqrt{{\displaystyle \frac{1}{5}}}}=\sqrt{5}c \end{gathered}$$

Therefore, the zeroth component of the 4-velocity is $\eta °=\sqrt{5}c$

Determine the ordinary velocity components υx ¯and υy¯ in S¯:

(d)

Write the x-component of the ordinary velocity in .

Substitute for and in the above expression.

Write the y-component of the ordinary velocity in .

……. (3)

Here, is the Lorentz contraction which is given by,

Substitute in the above expression.

Substitute for, for and for in equation (3).

Therefore, the components and of the ordinary velocity in are and respectively.

Determine the proper velocity components ηx¯ and ηy¯ in S¯:

(e)

Write the x-component of the proper velocity in .

$\overline{{\eta }_x}=\gamma \left({\eta }_x-\beta \eta °\right)$ …… (4)

Here, $\beta$is the Lorentz factor which is given by,

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

Substitute $\gamma =\sqrt{\frac{5}{3}},\sqrt{2}c,\beta =\frac{v}{c},\eta °=\sqrt{5}c$and $v=\sqrt{\frac{2}{5}}c$in equation (4).

$$\begin{gathered} \overline{{\eta }_x}=\sqrt{\frac{5}{3}}\left(\sqrt{2}c-\frac{v}{c}\sqrt{5}c\right) \\ \overline{{\eta }_x}=\sqrt{\frac{5}{3}}\left(\sqrt{2}c-\frac{\left(\sqrt{{\displaystyle \frac{2}{5}}}c\right)}{c}\sqrt{5}c\right) \\ \overline{{\eta }_x}=\sqrt{\frac{5}{3}}\left(\sqrt{2}c-\sqrt{2}c\right)=0 \end{gathered}$$

Write the y-component of the ordinary velocity in $\overline{S}$.

$\overline{{\eta }_x}=\overline{{\eta }_y}$

Substitute $\sqrt{2}c$for ${\eta }_y$in the above expression.

$\overline{{\eta }_y}=\sqrt{2}c$

Therefore, the components $\overline{{\eta }_x}$and $\overline{{\eta }_y}$of the proper velocity in $\overline{S}$are 0 and $\sqrt{2}c$respectively.

Verify the equation η¯=u¯1-u¯2c2

• Solve RHS value:

Write the value of .

$\overline{u}=\sqrt{{\overline{u_x}}^2+{\overline{u_y}}^2}$

Substitute 0 for $\overline{u_x}$and $\sqrt{\frac{2}{3}}c$for $\overline{u_y}$in the above equation.

$$\begin{gathered} \overline{u}=\sqrt{{\left(0\right)}^2+{\left(\sqrt{\frac{2}{3}}c\right)}^2} \\ =\sqrt{\frac{2}{3}c} \end{gathered}$$

Calculate the R.H.S value.

$\frac{\left(\sqrt{{\displaystyle \frac{2}{3}}}c\right)}{\sqrt{1-{\displaystyle \frac{{\left(\sqrt{{\displaystyle \frac{2}{3}}}c\right)}^2}{c^2}}}}=\frac{\left(\sqrt{{\displaystyle \frac{2}{3}}}c\right)}{\sqrt{1-\left({\displaystyle \frac{2}{3}}\right)}}=\left(\sqrt{\frac{2}{3}}c\right)\times \left(\sqrt{3}\right)=\sqrt{2}c$

• Solve L.H.S value:

Write the value of $\overline{\eta }$.

Substitute 0 for$\overline{{\eta }_x}$ and $\sqrt{2}c$for$\overline{{\eta }_y}$ in the above equation.

$$\begin{gathered} \overline{\eta }=\sqrt{{\left(0\right)}^2+{\left(\sqrt{2}c\right)}^2} \\ \overline{\eta }=\sqrt{2}c \end{gathered}$$

Hence, L.H.S = R.H.S

Hence verified.

Therefore, it is proved that $\overline{\eta }=\frac{\overline{u}}{\sqrt{1-{\displaystyle \frac{{\overline{u}}^2}{c^2}}}}$.