UM-StL 1D Acceleration Solver

Do you want to skip down to explore a physics challenge, like guessing the time until you hit the water after jumping off a 50 foot bridge, estimating how many g's of acceleration you'll feel in accelerating a car from 0 to 60 mph in 8 seconds, or figuring the initial velocity of a speeding car which left 30 "meters of rubber" on the road in a 2 second screeching stop? If as with these challenges your problem involves acceleration which is constant (or nearly constant) in magnitude AND direction, you may find some guidance here. If what follows loses you entirely, see L. Gladney's Notes on Constant Acceleration at the University of Pennsylvania for a more fundamental primer on such challenges first.

A novel feature of this page is that it also allows you to consider relativistic times and velocities, which make life more interesting as speeds approach that of light. For example, you might ask HOW FAR acceleration at one g for 50 years would take a traveler, assuming EITHER that the 50 years elapses on the clocks of an unaccelerated observer simply monitoring the expedition, OR that it elapses on the clocks of the accelerated traveler. Relativity has noticeable effects here: The two answers differ by nearly twenty powers of ten (cf. Lagoute & Davost 1995 in Am. J. Phys. 63, 221)! See also our Andromeda problem in "cold application testing" now.

You have several choices on how to proceed at this point. One choice is to run these calculations on OUR computer through your web browser. This opportunity is provided below on this page, at least at those times when our query server is up and running. Another choice is to do the calculations on YOUR computer using MathCAD, or the MathSoft Browser available free here. For that purpose, we have set up 4 MathCAD worksheets, one for Newtonian or Low-Velocity Inputs, one for Relativistic Coordinate (Non-Accelerated Observer) Inputs, one for Traveler-Kinematic or Proper-Time/4-Velocity Inputs, and the last for Mixed-Kinematic Inputs. See below (and our relativity table of contents) for clues to what these things might mean. Thirdly, you may download a Visual BASIC program to do the calculations independently, from here. Lastly, you might want to try graphical solution of a problem. The picture below points to a page under development toward this goal.

To run OUR query server, which for now solves relativistic problems exactly only from Newtonian inputs, first decide WHICH of the 5 constant acceleration variables you already know. These variables are the acceleration itself (a₀), the distance traveled (dx), the time elapsed (e.g. dt), and the initial/final velocities (e.g. v₀/v_f). You need to know three of them to solve a problem uniquely, as in these ordinary speed and relativistic examples.

When filling out the forms below you will soon be able to switch from the default Galilean or "Newtonian" ( low-velocity approximation) time and velocity setup, to setups in terms of relativistic "coordinate" (unaccelerated observer) or "proper" (traveler-kinematic) times and speeds. These connections are discussed in the papers gr-qc/9512012, gr-qc/9607038, and physics/9611011. To retain clarity with 3 self-consistent kinematics, we use t and v for Galilean time and velocity, b and w for coordinate time and velocity, and T (or tau) and u for the traveler's proper time and "spatial 4- velocity", respectively. Be cautious here, since this notation changes with context from place to place.

We often quote inertial coordinate times in [iyears], traveler or proper times in [tyears], coordinate velocities in units of [c] (the speed of light in [lightyears/iyear]), and proper velocities in [lightyears/tyear] or "roddenberries" [rb]. The latter name seems to have mnemonic value to students, perhaps via its mental connections to "hotrod" and the producer of the first Star Trek series which, like proper velocity, ignores the lightspeed limit to which coordinate velocity is held. Deriving the inter-relationship between these parameters is simple (in html or postscript). Regardless of how you set up the problem, the inferred value of relativistic AND Newtonian variables (11 variables in all) will be provided in the Results Report.

Note: These routines will work for solving constant acceleration problems in more than one spatial dimension, by simply ignoring that component of the velocity which is perpendicular to the direction of constant acceleration, if and only if that perpendicular velocity component is much less than the speed of light! This server might in the future be extended to include significant values of this "v_perp", although under such conditions only the two relativistic kinematics retain precise self-consistency.

Accel-One's Browser-Interactive Setup: Select 3 out of the 5 variables

1. distance traveled dx.
2. time elapsed e.g. dt.
3. initial velocity e.g. v₀.
4. final velocity e.g. v_f.
5. constant acceleration a₀.

Setup Kinematic (Times & Velocities) to Use -- Only "Newtonian" for Now:

1. Galilean or "Newtonian" time/velocity
2. relativistic coordinate time/velocity
3. proper or traveler-kinematic time/velocity

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