This problem is *part only* of a problem found in *Fundamentals of Physics*, Chapter 2, Page 30, 4th Edition (John Wiley & Sons, 1993) by Halliday, Resnick, and Walker. As we suggest below, the remainder of the problem (not included here) is definitely worth looking up!

The **Results Report** will then read:

```
```

```
Distance Traveled: dx = -260 [m]
```

`Initial Newtonian Velocity: v`_{o} = 0 [m/sec]

`Constant Proper Acceleration: a`_{o} = -9.8 [m/sec^2]

*Newtonian (Low-Velocity Kinematic) Results*:```
```

`Final Newtonian Velocity: v`_{f} = {71.38, -71.38} [m/sec]

`Newtonian Time Elapsed: dt = {-7.28, +7.28} [sec]`

These values are obtained from the usual Newtonian constant acceleration equations, using v_{f}=(+/-)SQRT[v_{o}^2+2a_{o}dx] and dt=[v_{f}-v_{o}]/a_{o}. The actual constant acceleration problem is now solved, and only routine conversion of velocities and elapsed time to the two relativistic kinematics remains.

*Accelerated Traveler (Proper Time) Results*:```
```

`Initial Spatial 4-Velocity: u`_{o} = 0 [rb or lightyears/tyear]

`Final Spatial 4-Velocity: u`_{f} = {2.38x10^-7, -2.38x10^-7} [rb or lightyears/tyear]

`Traveler (Proper) Time Elapsed: dT = {-2.38x10^-7,+2.38x10^-7} [tyears]`

These traveler-frame results follow using a routine protocol for converting from values for the Newtonian kinematic (whose time-parameterization corresponds to no-one's clocks at high velocity), via u=v*SQRT[1+0.25*(v/c)^2] and dT=[ASINH(u_{f}/c)-ASINH(u_{o}/c)]*c/a_{o} where c=3x10^8 m/s is the speed of light. In this particular problem, the traveler values are essentially the same as the Newtonian values.

*Relativistic Inertial-Frame Results*:```
```

`Starting Inertial Velocity: w`_{o} = 0 [c or lightyears/iyear]

`Final Inertial Velocity: w`_{f} = {2.38x10^-7, -2.38x10^-7} [c or lightyears/iyear]

`Inertial Time Elapsed: db = {-2.38x10^-7,+2.38x10^-7} [iyears]`

These inertial-frame results follow by routine conversion from the traveler-frame values, using w=u/SQRT[1+(u/c)^2], and db=[u_{f}-u_{o}]/a_{o}. In this particular problem, the inertial values are essentially the same as the Newtonian values.

This same problem can be solved graphically with a general-purpose (1 for all) or universal x-tv diagram, as discussed elsewhere and shown below. Note that in this non-relativistic parameter range (i.e. where each variable is much less than one when put in dimensionless form), all three (i.e. Newtonian, traveler, & inertial) time variables AND all three velocity variables plot onto *a single curve*!

Note: This graph is simply a (big time!) "zoomed-in" image near the *origin* of the plot used for our relativistic example. That's because everyday speeds are small compared to the speed of light. For example, 55 mph is only 82 nano-roddenberries! This wide "dynamic range" for the constant acceleration problems one *might* try to solve with such a graph is one reason we went to log-log axes on the nomograph page. Ordinary plots, like the one above, have their advantages as well, and either can be used to visualize the variables in *any* constant acceleration problem.

P.S. If you echo this material in print or on web, a citation would be cool. `{Thanks. :) /pf}`

Send suggestions and comments to philfSPAMBLOCK@newton.umsl.edu. Mindquilts site page requests ~2000/day approaching a million per year. Requests for a "stat-counter linked subset of pages" since 4/7/2005: .