The primed-frame axes in the transformation diagram of Sample Problem 1 are easy to locate for any relative frame velocity V/c, because on the orthogonal x, ct coordinate diagram, the ct' axis corresponds to the line x = (V/c)ct and the x' axis corresponds to the line ct = (V/c)x. It is trickier to measure locations x' and times ct' on the diagrams, however, because the primed coordinate axes are nonorthogonal. One way to calibrate these primed coordinates is to draw "invariant hyperbolas" on the diagram, which are the curves -(ct')² + x^² = −(c1)² + x² = ta², where a = some constant length. For example, if we choose a = 1 meter and the plus sign, this curve intersects the x axis (where ct = : 0) at x = 1 m, and it also intersects the x' axis (where ct' =0) at x' = 1 m, as shown on the next page. To find the x, ct coordinates of this latter point, note that the x' axis corresponds to the line ct = (V/c)x, so eliminating we have -(V/c)²x² + x² = (1 m)2, from which we find that this hyperbola intersects the x' axis at = y.lm 3- x = in terms of the usual gamma factor between the two frames. Similarly, the curve with a = 1 m and the minus sign intersects the ct axis at ct = 1 m and the ct' axis at ct' = 1 m. This hyperbola is also drawn. (a) Show that all the above-mentioned hyperbolas asymptotically approach the line x = ct (or x = -ct) for large valucs of any one of the four coordinates. ct (m) 2 Im √I-V²/C² ct' (m) T 123 -x' (m) T -x (m)
The primed-frame axes in the transformation diagram of Sample Problem 1 are easy to locate for any relative frame velocity V/c, because on the orthogonal x, ct coordinate diagram, the ct' axis corresponds to the line x = (V/c)ct and the x' axis corresponds to the line ct = (V/c)x. It is trickier to measure locations x' and times ct' on the diagrams, however, because the primed coordinate axes are nonorthogonal. One way to calibrate these primed coordinates is to draw "invariant hyperbolas" on the diagram, which are the curves -(ct')² + x^² = −(c1)² + x² = ta², where a = some constant length. For example, if we choose a = 1 meter and the plus sign, this curve intersects the x axis (where ct = : 0) at x = 1 m, and it also intersects the x' axis (where ct' =0) at x' = 1 m, as shown on the next page. To find the x, ct coordinates of this latter point, note that the x' axis corresponds to the line ct = (V/c)x, so eliminating we have -(V/c)²x² + x² = (1 m)2, from which we find that this hyperbola intersects the x' axis at = y.lm 3- x = in terms of the usual gamma factor between the two frames. Similarly, the curve with a = 1 m and the minus sign intersects the ct axis at ct = 1 m and the ct' axis at ct' = 1 m. This hyperbola is also drawn. (a) Show that all the above-mentioned hyperbolas asymptotically approach the line x = ct (or x = -ct) for large valucs of any one of the four coordinates. ct (m) 2 Im √I-V²/C² ct' (m) T 123 -x' (m) T -x (m)
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From the provided information, show that all of the mentioned hyperboals asmptotically approach the line x = ct for large values of any one of the four mentioned coordinates.
Transcribed Image Text:The primed-frame axes in the transformation diagram of Sample Problem 1
are easy to locate for any relative frame velocity V/c, because on the orthogonal
x, ct coordinate diagram, the ct' axis corresponds to the line x = (V/c)ct and the
x' axis corresponds to the line ct = (V/c)x. It is trickier to measure locations x'
and times ct' on the diagrams, however, because the primed coordinate axes are
nonorthogonal. One way to calibrate these primed coordinates is to draw "invariant
hyperbolas" on the diagram, which are the curves -(ct')² + x^² = −(ct)² + x² =
ta², where a = some constant length.
For example, if we
choose a = 1 meter and the plus sign, this curve intersects the x axis (where ct = 0)
at x = 1 m, and it also intersects the x' axis (where ct' = 0) at x' = 1 m, as shown on
the next page. To find the x, ct coordinates of this latter point, note that the x' axis
corresponds to the line ct = (V/c)x, so eliminating we have -(V/c)²x² + x² =
(1 m)², from which we find that this hyperbola intersects the x' axis at
=y.lm
in terms of the usual gamma factor between the two frames. Similarly, the curve
with a = 1 m and the minus sign intersects the ct axis at ct = 1 m and the ct' axis
at ct' = 1 m. This hyperbola is also drawn. (a) Show that all the above-mentioned
hyperbolas asymptotically approach the line x = ct (or x = −ct) for large valucs
of any one of the four coordinates.
ct (m)
4
3-
2
X =
Im
/1-V²/C²
ct' (m)
2
3
-x' (m)
-x (m)
Transcribed Image Text:Sample Problem 1 transformation diagram
ct
ct'
·X
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