File:Constantaccelerationtrips.png

This book^[1] like most intro physics books lists a set of 3 or 4 constant acceleration equations. The most compact way to summarize these equations may be as follows:

${\displaystyle a\equiv {\frac {\Delta v}{\Delta t}}={\frac {{\frac {1}{2}}\Delta (v^{2})}{\Delta x}}={\frac {\Delta x-v_{o}\Delta t}{{\frac {1}{2}}(\Delta t)^{2}}}}$.

Here by definition the Greek letter delta (Δ) preceeding any variable is taken to mean final value minus initial value, i.e. Δz ≡ z_final-z_initial.

The time elapsed on a "four-leg" constant-acceleration round trip traveling a distance x_target and back is therefore:

${\displaystyle t_{\text{roundtrip}}=4{\sqrt {\frac {x_{\text{target}}}{a}}}=4\left({\frac {v_{\text{max}}}{a}}\right)}$.

Here the maximum speed on this trip is:

${\displaystyle v_{\text{max}}=a\left({\frac {t_{\text{roundtrip}}}{4}}\right)={\sqrt {ax_{\text{target}}}}}$.

The target distance can then be written:

${\displaystyle x_{\text{target}}=a\left({\frac {t_{\text{roundtrip}}}{4}}\right)^{2}={\frac {v_{\text{max}}^{2}}{a}}}$.

It might also be interesting to ask^[2] what anyspeed versions of these 6 equalities look like.

As suggested by the table below, the anyspeed versions are more complicated, in part because differences between elapsed map-times (coordinate-time t) and traveler-times (proper-time τ) at high speed (v ~ c) give rise to a distinction between coordinate versus proper velocities (v≡dx/dt versus w≡dx/dτ) and coordinate versus proper accelerations (a versus α) as well as coordinate versus proper times.

At high speeds as one moves from the variables: time, through velocity (change in map-distance per unit time), to acceleration (change in velocity per unit time) the relevance of the "coordinate-versions" of these quantities becomes increasingly parochial (i.e. frame-specific) in comparison to their "proper" analogs (which address matters about the traveler's motion likely to be of more global interest). Hence in the table we haven't bothered to mention either maximum coordinate-speed v_max or the extremes of coordinate-acceleration a, as they are of both limited interest and even messier to write out.

constant proper-acceleration roundtrips: the approximation ≈ works when v_max ≈ w_max << lightspeed c
e.g. at which point proper-acceleration α ≈ coordinate-acceleration a and proper-velocity w ≈ coordinate-velocity v

⇓get \ given⇒	target distance ${\displaystyle x\,}$	proper-speed max ${\displaystyle w_{\text{max}}\simeq v_{\text{max}}}$	return trav-time ${\displaystyle \tau \simeq t}$	return map-time ${\displaystyle t\,}$
target distance ${\displaystyle x\,}$	${\displaystyle x\,}$	${\displaystyle 2{\tfrac {c^{2}}{\alpha }}({\sqrt {1+({\tfrac {w_{\text{max}}}{c}})^{2}}}-1)}$ ${\displaystyle \simeq {\tfrac {v_{\text{max}}^{2}}{a}}}$	${\displaystyle 2{\tfrac {c^{2}}{\alpha }}(\cosh \left[{\tfrac {\alpha }{c}}{\tfrac {\tau }{4}}\right]-1)}$ ${\displaystyle \simeq a\left({\tfrac {t}{4}}\right)^{2}}$	${\displaystyle 2{\tfrac {c^{2}}{\alpha }}({\sqrt {1+({\tfrac {\alpha }{c}}{\tfrac {t}{4}})^{2}}}-1)}$ ${\displaystyle \simeq a({\tfrac {t}{4}})^{2}}$
proper-speed max ${\displaystyle w_{\text{max}}}$	${\displaystyle {\sqrt {\alpha x(1+{\tfrac {\alpha x}{4c^{2}}})}}}$ ${\displaystyle \simeq {\sqrt {ax}}}$	${\displaystyle w_{\text{max}}\,}$ ${\displaystyle \simeq v_{\text{max}}}$	${\displaystyle c\sinh \left[{\tfrac {\alpha }{c}}{\tfrac {\tau }{4}}\right]}$ ${\displaystyle \simeq a{\tfrac {t}{4}}}$	${\displaystyle \alpha {\tfrac {t}{4}}}$ ${\displaystyle \simeq a{\tfrac {t}{4}}}$
return trav-time ${\displaystyle \tau \,}$	${\displaystyle 4{\tfrac {c}{\alpha }}\cosh ^{-1}\left[1+{\tfrac {\alpha }{c^{2}}}{\tfrac {x}{2}}\right]}$ ${\displaystyle \simeq 4{\sqrt {\tfrac {x}{a}}}}$	${\displaystyle 4{\tfrac {c}{\alpha }}\sinh ^{-1}\left[{\tfrac {w_{\text{max}}}{c}}\right]}$ ${\displaystyle \simeq 4{\tfrac {v_{\text{max}}}{a}}}$	${\displaystyle \tau \,}$ ${\displaystyle \simeq t}$	${\displaystyle 4{\tfrac {c}{\alpha }}\sinh ^{-1}\left[{\tfrac {\alpha }{c}}{\tfrac {t}{4}}\right]}$ ${\displaystyle \simeq t}$
return map-time ${\displaystyle t\,}$	${\displaystyle 4{\sqrt {{\tfrac {x}{\alpha }}(1+{\tfrac {\alpha x}{4c^{2}}})}}}$ ${\displaystyle \simeq 4{\sqrt {\tfrac {x}{a}}}}$	${\displaystyle 4{\tfrac {w_{\text{max}}}{\alpha }}}$ ${\displaystyle \simeq 4{\tfrac {v_{\text{max}}}{a}}}$	${\displaystyle 4{\tfrac {c}{\alpha }}\sinh \left[{\tfrac {\alpha }{c}}{\tfrac {\tau }{4}}\right]}$ ${\displaystyle \simeq t}$	${\displaystyle t\,}$

1. ↑ Raymond A. Serway and John W. Jewett (2004 6th ed) Physics for Scientists and Engineers (Brooks/Cole - Thomson Learning, Belmont CA).
2. ↑ P. Fraundorf (2012) "A fun intro to 1D kinematics", arXiv:1206.2877 [physics.pop-ph].

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