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| ### Statement |
| ### Statement |
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| $14.2.14.$ [Insert the problem statement] |
| $14.2.14.$ |
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| The Lorentz transformation makes it possible to know what will happen if |
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| we observe a phenomenon moving relative to the object, the carrier of the |
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| phenomenon, at a speed v, or if the object moves relative to us at a speed v, |
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| provided that we know how the phenomenon occurs when the object is sta- |
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| tionary. Therefore, the following statement of the question will often be used |
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| in the future. In a stationary system, the phenomenon is described. How |
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| will this phenomenon occur if the object that carries the phenomenon is mov- |
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| ing at speed v? The answer assumes a description of this phenomenon in the |
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| system a reference point that moves at a speed of −v relative to the system |
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| in which the phenomenon is described. This is equivalent to describing this |
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| phenomenon in the case of the movement of an object, the carrier of the phe- |
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| nomenon, with a velocity v relative to a stationary observer. The second option |
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| is interesting because it can be extended to several isolated objects moving at |
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| different speeds. Use this to solve the following problem. The observation sta- |
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| tion recorded light signals from two rockets moving in a straight line towards |
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| the station. The frequencies of signals registered by the station are v1and v2. |
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| The signal frequency of stationary rockets is equal to v0. How fast do rockets |
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| approach each other? |
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| ### Solution |
| ### Solution |
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| Velocity of each rocket from the station |
| Velocity of each rocket from the station |
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| The station receives frequencies $\nu_1$ and $\nu_2 $from sources that are approaching. The Doppler effect formula for direct approach$ (\beta = v/c) $is: | | The station receives frequencies $\nu_1$ and $\nu_2 $from sources that are approaching. The Doppler effect formula for direct approach$ (\beta = v/c) $is: |
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| $\nu_i = \nu_0 \sqrt{\frac{1+\beta_i}{1-\beta_i}}, \qquad i=1,2$ | | $\nu_i = \nu_0 \sqrt{\frac{1+\beta_i}{1-\beta_i}}, \qquad i=1,2$ |
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| Solving for $\beta_i$ | | Solving for $\beta_i$ |
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| $\left(\frac{\nu_i}{\nu_0}\right)^2 = \frac{1+\beta_i}{1-\beta_i} | | $\left(\frac{\nu_i}{\nu_0}\right)^2 = \frac{1+\beta_i}{1-\beta_i} |
| \quad\Longrightarrow\quad | | \quad\Longrightarrow\quad |
| \beta_i = \frac{(\nu_i/\nu_0)^2 - 1}{(\nu_i/\nu_0)^2 + 1}$ | | \beta_i = \frac{(\nu_i/\nu_0)^2 - 1}{(\nu_i/\nu_0)^2 + 1}$ |
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| In terms of velocity: | | In terms of velocity: |
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| $v_i = c\,\frac{\nu_i^2 - \nu_0^2}{\nu_i^2 + \nu_0^2}$ | | $v_i = c\,\frac{\nu_i^2 - \nu_0^2}{\nu_i^2 + \nu_0^2}$ |
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| Velocity difference in the station's frame | | Velocity difference in the station's frame |
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| The rockets move along the same line toward the station. The speed with which they approach each other, measured in the station's frame, is simply the difference of their velocities (if one is faster than the other) or the sum (if they come from opposite directions). We assume they move in the same direction (one behind the other): | | The rockets move along the same line toward the station. The speed with which they approach each other, measured in the station's frame, is simply the difference of their velocities (if one is faster than the other) or the sum (if they come from opposite directions). We assume they move in the same direction (one behind the other): |
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| $\Delta v = v_1 - v_2$ | | $\Delta v = v_1 - v_2$ |
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| Substituting the expressions for $v_1$ and $v_2$ | | Substituting the expressions for $v_1$ and $v_2$ |
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| $\Delta v = c\left( \frac{\nu_1^2 - \nu_0^2}{\nu_1^2 + \nu_0^2} - \frac{\nu_2^2 - \nu_0^2}{\nu_2^2 + \nu_0^2} \right)$ | | $\Delta v = c\left( \frac{\nu_1^2 - \nu_0^2}{\nu_1^2 + \nu_0^2} - \frac{\nu_2^2 - \nu_0^2}{\nu_2^2 + \nu_0^2} \right)$ |
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| Simplification | | Simplification |
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| We put a common denominator: | | We put a common denominator: |
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| $\Delta v = c\,\frac{(\nu_1^2 - \nu_0^2)(\nu_2^2 + \nu_0^2) - (\nu_2^2 - \nu_0^2)(\nu_1^2 + \nu_0^2)}{(\nu_1^2 + \nu_0^2)(\nu_2^2 + \nu_0^2)}$ | | $\Delta v = c\,\frac{(\nu_1^2 - \nu_0^2)(\nu_2^2 + \nu_0^2) - (\nu_2^2 - \nu_0^2)(\nu_1^2 + \nu_0^2)}{(\nu_1^2 + \nu_0^2)(\nu_2^2 + \nu_0^2)}$ |
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| Expanding the numerator: | | Expanding the numerator: |
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| $\begin{aligned} | | $\begin{aligned} |
| &(\nu_1^2 - \nu_0^2)(\nu_2^2 + \nu_0^2) - (\nu_2^2 - \nu_0^2)(\nu_1^2 + \nu_0^2) \\ | | &(\nu_1^2 - \nu_0^2)(\nu_2^2 + \nu_0^2) - (\nu_2^2 - \nu_0^2)(\nu_1^2 + \nu_0^2) \\ |
| &= (\nu_1^2\nu_2^2 + \nu_1^2\nu_0^2 - \nu_0^2\nu_2^2 - \nu_0^4) - (\nu_1^2\nu_2^2 + \nu_2^2\nu_0^2 - \nu_0^2\nu_1^2 - \nu_0^4) \\ | | &= (\nu_1^2\nu_2^2 + \nu_1^2\nu_0^2 - \nu_0^2\nu_2^2 - \nu_0^4) - (\nu_1^2\nu_2^2 + \nu_2^2\nu_0^2 - \nu_0^2\nu_1^2 - \nu_0^4) \\ |
| &= \cancel{\nu_1^2\nu_2^2} + \nu_1^2\nu_0^2 - \nu_0^2\nu_2^2 - \nu_0^4 - \cancel{\nu_1^2\nu_2^2} - \nu_2^2\nu_0^2 + \nu_0^2\nu_1^2 + \nu_0^4 \\ | | &= \cancel{\nu_1^2\nu_2^2} + \nu_1^2\nu_0^2 - \nu_0^2\nu_2^2 - \nu_0^4 - \cancel{\nu_1^2\nu_2^2} - \nu_2^2\nu_0^2 + \nu_0^2\nu_1^2 + \nu_0^4 \\ |
| &= 2\nu_0^2(\nu_1^2 - \nu_2^2). | | &= 2\nu_0^2(\nu_1^2 - \nu_2^2). |
| \end{aligned}$ | | \end{aligned}$ |
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| The denominator is: | | The denominator is: |
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| $(\nu_1^2 + \nu_0^2)(\nu_2^2 + \nu_0^2)$ | | $(\nu_1^2 + \nu_0^2)(\nu_2^2 + \nu_0^2)$ |
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| Therefore: | | Therefore: |
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| $\Delta v = c\,\frac{2\nu_0^2(\nu_1^2 - \nu_2^2)}{(\nu_1^2 + \nu_0^2)(\nu_2^2 + \nu_0^2)}$ | | $\Delta v = c\,\frac{2\nu_0^2(\nu_1^2 - \nu_2^2)}{(\nu_1^2 + \nu_0^2)(\nu_2^2 + \nu_0^2)}$ |
