Project plate displacement with explicit assumptions

Turn a rate into a conditional displacement

A velocity answers how position changes per unit time. A displacement answers how much the position changes over a chosen interval. Multiplying the two is appropriate when the velocity is constant. In this investigation, the velocities and local axes are held fixed for the calculation; that assumption is the reason the component graphs are straight lines.

Start at across 3 and along 4 cm/year. Over ten years, B’s displacement relative to A is 30 cm east and 40 cm north. Its magnitude is 50 cm. These are changes in relative position, not the absolute coordinates of either marker and not an initial distance between them. Check the ten-year numbers before attempting the much longer interval.

Convert the units in visible steps

Myr means one million years. A kilometer contains 100,000 centimeters. Therefore a rate of 1 cm/year acting for 1 Myr gives 1,000,000 cm, or 10 km. Equivalently, 1 cm/year = 10 km/Myr. The larger numerical value in kilometers per million years comes from the much longer time unit, not a change in physical speed.

For five million years, multiply each control by 10 km/Myr per cm/year and then by 5 Myr. The east displacement is 50a km and the north displacement is 50b km, where a and b are the signed control values in cm/year. At the default, these are 150 and 200 km. Their magnitude is √(150² + 200²) = 250 km.

Write the units beside every line of one calculation. A useful check is that years cancel against years and centimeters convert to kilometers. A result still carrying “per year” is a velocity, not the requested displacement. Keep signs: a negative projected component means motion toward west or south in these axes.

Read the graph as a model calculation

At zero elapsed time the projected displacement components are both zero. At one Myr, the default components are 30 and 40 km; at five Myr they are 150 and 200 km. Read an intermediate point, calculate it independently, and compare with the endpoint trend. The line’s slope represents a constant component velocity expressed in km/Myr.

Change across to −3 while keeping along at 4. The east component now decreases below zero while the north component increases. The magnitude after five Myr is still 250 km. This comparison distinguishes signed component graphs from a nonnegative magnitude. A graph below zero is not a negative distance traveled; it records direction in the chosen coordinate system.

Draw the endpoint vector on paper with a stated scale, such as one square per 50 km. Label its east and north projections separately from its diagonal length. Use the same scale in both directions so the geometry does not distort the component comparison.

Separate relative displacement from a landscape forecast

The default normal component contributes 150 km to the projected relative displacement; the full vector magnitude is 250 km. Neither is automatically a future ocean width. No initial width, moving coastline, spreading-center migration, deforming boundary zone or spherical reconstruction has been supplied. The model is a local planar extrapolation of two velocities, not a geographic evolution simulator.

Likewise, a convergent component is not a calculated amount of rock consumed by subduction. That inference would require a compatible geological setting and additional geometry. A precise arithmetic result can be conditional on assumptions that are too limited for the broader question.

Present-day satellite measurements and geological records sample different timescales. Ocean-floor magnetic reversal records can support estimates of average motion over geological intervals, whereas repeated GPS observations constrain recent motion. Comparing such evidence can help test whether a short-term rate is a suitable long-term approximation. USGS discussion of plate-motion measurements.

Design an evidence check

Make two columns headed “calculated from assumptions” and “needed to assess the projection.” Put the five-Myr displacement in the first. In the second, include a documented rate estimate, its reference frame and interval, uncertainty, and independent evidence about past motion. Explain why repeated outputs from unchanged slider settings are not independent observations supporting a constant rate.

The five-Myr display is an illustrative extrapolation. It does not assign probability to the future path, estimate earthquake recurrence or show measured motion over that whole interval. State those limits in the same paragraph as your numerical conclusion.

Transfer to a different interval

A paper-only case keeps relative velocity at (−2, 5) cm/year for 2 Myr. Calculate the two displacement components and their magnitude. Which component describes normal convergence in the supplied north-south boundary geometry?

Check the plate-projection transfer

The projected components are (−40, 100) km. Their magnitude is √11600, approximately 107.7 km. The −40 km east component represents westward relative displacement and normal convergence in this geometry. The 107.7 km magnitude includes the along-boundary part; it is not an ocean-width change or a prediction of subducted material.