# Example Problem for the Accel-1D Solver

Problem: In 1938, Joe Sprinz of the San Francisco Baseball Club attempted to break the record for catching a baseball dropped from the greatest height. Members of the Cleveland Indians had set the record the preceeding year when they caught baseballs dropped about 230 meters from atop a building. Sprinz used a blimp at 260 meters. Ignore the effects of air on the ball. Find the time of its fall, and the speed of its impact.

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!

• We now prefer a more standardized notation, but for the time being...
• Notation Key B (used on this page) is: Galilean {time, velocity}={t, v}, coordinate {time, velocity}={b, w}, and proper {time, velocity}={T, u}.
• URL: http://newton.umsl.edu/infophys/a1exampl.html
• Last modified on 03 Dec 1996.
• Dept. of Physics & Astronomy, University of Missouri-StL, St. Louis MO 63121-4499
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Method for Solving: On the opening form, select as knowns the three variables: distance traveled dx, acceleration ao, and initial velocity (e.g. vo). Also select the Newtonian kinematic, since the velocities are unlikely to be relativistic. When prompted on the second form, then specify: dx = 260 m, ao = 1g = 9.8 m/s^2, and vo=0 m/s.

The Results Report will then read:

Setup: You entered the following:``` ```
• ```Distance Traveled: dx = -260 [m] ```
• ```Initial Newtonian Velocity: vo = 0 [m/sec] ```
• `Constant Proper Acceleration: ao = -9.8 [m/sec^2]`

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

• ```Final Newtonian Velocity: vf = {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 vf=(+/-)SQRT[vo^2+2aodx] and dt=[vf-vo]/ao. 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: uo = 0 [rb or lightyears/tyear] ```
• ```Final Spatial 4-Velocity: uf = {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(uf/c)-ASINH(uo/c)]*c/ao 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: wo = 0 [c or lightyears/iyear] ```
• ```Final Inertial Velocity: wf = {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=[uf-uo]/ao. In this particular problem, the inertial values are essentially the same as the Newtonian values.

Final Discussion: You were presented with two possible solutions -- choose the one for which vf is negative (downward). The other solution would yield the velocity of the ball at the ground if it had been thrown upward to the blimp prior to its fall. The Newtonian time elapsed is dt = 7.28 seconds, and the final velocity is vf = -71.38 meters per second. That is -159 miles per hour. Not a slow pitch! As predicted for velocities small compared to the speed of light, both relativistic inertial time db and the relativistic proper time dT agree with the Newtonian values, as do the relativistic velocities w and u. To find out what happened to Joe after he got his glove on a ball (after 5 tries), look at a copy of Fundamentals of Physics, Chapter 2, Page 30, 4th Edition (John Wiley & Sons, 1993) by Halliday, Resnick, and Walker.

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}`

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