Mean speed theorem explained

Mean speed theorem should not be confused with The Merton Rule.

The mean speed theorem, also known as the Merton rule of uniform acceleration,^[1] was discovered in the 14th century by the Oxford Calculators of Merton College, and was proved by Nicole Oresme. It states that a uniformly accelerated body (starting from rest, i.e. zero initial velocity) travels the same distance as a body with uniform speed whose speed is half the final velocity of the accelerated body.^[2]

Details

Oresme provided a geometrical verification for the generalized Merton rule, which we would express today as

s = (1)/(2) (v _{0} + v _{\rm} )t (i.e., distance traveled is equal to one half of the sum of the initial v _{0} and final v _{\rm} velocities, multiplied by the elapsed time t ), by finding the area of a trapezoid.^[3] Clay tablets used in Babylonian astronomy (350–50 BC) present trapezoid procedures for computing Jupiter's position and motion and anticipate the theorem by 14 centuries.^[4]

The medieval scientists demonstrated this theorem—the foundation of "the law of falling bodies"—long before Galileo, who is generally credited with it. Oresme's proof is also the first known example of the modelization of a physical problem as a mathematical function with a graphical representation, as well as of an early form of integration, thus laying the foundation of calculus. The mathematical physicist and historian of science Clifford Truesdell, wrote:^[5]

The theorem is a special case of the more general kinematics equations for uniform acceleration.

See also

• Science in the Middle Ages
• Scholasticism

Further reading

• Sylla, Edith (1982) "The Oxford Calculators", in Kretzmann, Kenny & Pinborg (edd.), The Cambridge History of Later Medieval Philosophy.
• Longeway, John (2003) "William Heytesbury", in The Stanford Encyclopedia of Philosophy.

Notes and References

1. [Edward Grant]
2. Book: Boyer, Carl B. . Carl Benjamin Boyer . A History of the Calculus and Its Conceptual Development . Dover . 1959 . 978-0-486-60509-8 . https://books.google.com/books?id=KLQSHUW8FnUC&pg=PA79 . III. Medieval Contributions . 79–89 .
3. C. H. Edwards, Jr., The Historical Development of the Calculus (1979) pp. 88-89.
4. Ossendrijver . Mathieu . 29 Jan 2016 . Ancient Babylonian astronomers calculated Jupiter's position from the area under a time-velocity graph . Science . 351 . 6272 . 482–484 . 10.1126/science.aad8085 . 2016Sci...351..482O . 26823423. 206644971 .
5. Clifford Truesdell, Essays in The History of Mechanics, (Springer-Verlag, New York, 1968), p. 30