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Since the velocity of the object is the derivative of the position graph, the area under the line in the velocity vs. time graph is the displacement of the object. (Velocity is on the y-axis and time on the x-axis. Multiplying the velocity by the time, the time cancels out, and only displacement remains.)
Snap, [6] or jounce, [2] is the fourth derivative of the position vector with respect to time, or the rate of change of the jerk with respect to time. [4] Equivalently, it is the second derivative of acceleration or the third derivative of velocity, and is defined by any of the following equivalent expressions: = ȷ = = =.
The kinetic energy is , and since the particle is constrained to move along a curve, its velocity is simply /, where is the distance measured along the curve. Likewise, the gravitational potential energy gained in falling from an initial height y 0 {\displaystyle y_{0}} to a height y {\displaystyle y} is m g ( y 0 − y ) {\displaystyle mg(y_{0 ...
The curve of fastest descent is not a straight or polygonal line (blue) but a cycloid (red).. In physics and mathematics, a brachistochrone curve (from Ancient Greek βράχιστος χρόνος (brákhistos khrónos) 'shortest time'), [1] or curve of fastest descent, is the one lying on the plane between a point A and a lower point B, where B is not directly below A, on which a bead slides ...
The four-velocity at any point of world line () is defined as: = where is the four-position and is the proper time. [1] The four-velocity defined here using the proper time of an object does not exist for world lines for massless objects such as photons travelling at the speed of light; nor is it defined for tachyonic world lines, where the ...
Equation [3] involves the average velocity v + v 0 / 2 . Intuitively, the velocity increases linearly, so the average velocity multiplied by time is the distance traveled while increasing the velocity from v 0 to v, as can be illustrated graphically by plotting velocity against time as a straight line graph. Algebraically, it follows ...
A simple combinatorial rule for defining the winding number was proposed by August Ferdinand Möbius in 1865 [1] and again independently by James Waddell Alexander II in 1928. [2] Any curve partitions the plane into several connected regions, one of which is unbounded. The winding numbers of the curve around two points in the same region are equal.
The situation is depicted in the symmetric Loedel diagrams of Fig 4-1. Note that we can compare spacetime lengths on page directly with each other, due to the symmetric nature of the Loedel diagram. In Fig 4-2, the observer whose reference frame is given by the black axes is assumed to move from the origin O towards A.