Download
1 / 8
80 likes | 188 Vues
This piece explores Sir Isaac Newton's work on tangents and the Brachistochrone problem, a fascinating topic in physics and calculus. Newton proposed a solution involving the cycloid—a curve described by a point on a rolling circle. He showed that a weight falling from point A to point B follows this cycloidal path for the quickest descent under gravity. We delve into Newton's approach to finding tangents to curves and the implications of his findings on particle motion and velocity, illustrating a fundamental aspect of classical mechanics.
E N D
Newton’s Answer • From the given point A let there be drawn an unlimited straight line APCZ parallel to the horizontal, and on it let there be described an arbitrary cycloid AQP meeting the straight line AB (assumed drawn and produced if necessary) in the point Q, and further a second cycloid ADC whose base and height are to the base and height of the former as AB is to AQ respectively. This last cycloid will pass through the point B, and it will be that curve along which a weight, by the force of its gravity, shall descend most swiftly from the point A to the point B.
Newton’s Answer • In English : The cycloid. • What is a cycloid. • What Newton found. C:\Users\vincenzopie\Documents\PGCE\Newton Project\the inverted cycloid.gif http://archives.math.utk.edu/visual.calculus/0/parametric.5/cycloid.html
Newton and the motion of particles • Newton wanted to find his own method to find a tangent of a curve at a point. He did this by looking at a curve as the path a moving particle take. • http://www.fearofphysics.com/Proj/projectile.html In what directions does this ball travel?
Finding the Gradient using velocities. • We can call these velocities as “the change” in x and y over time. • The velocity vector of a particle lies on the tangent line at that point. • The common notation for this is. • How can we find the gradient using these quantities.
