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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 = = =.
Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk. See motion graphs and derivatives. A large number of fundamental equations in physics involve first or second time derivatives of quantities. Many other fundamental quantities in science are time derivatives of one another:
The last expression is the second derivative of position (x) with respect to time. On the graph of a function, the second derivative corresponds to the curvature or concavity of the graph. The graph of a function with a positive second derivative is upwardly concave, while the graph of a function with a negative second derivative curves in the ...
Further time derivatives have also been named, as snap or jounce (fourth derivative), crackle (fifth derivative), and pop (sixth derivative). [ 12 ] [ 13 ] The seventh derivative is known as "Bang," as it is a logical continuation to the cycle.
Since acceleration differentiates the expression involving position, it can be rewritten as a second derivative with respect to time: a = d 2 s d t 2 . {\displaystyle a={\frac {d^{2}s}{dt^{2}}}.} Since, for the purposes of mechanics such as this, integration is the opposite of differentiation, it is also possible to express position as a ...
As acceleration is defined as the derivative of velocity, v, with respect to time t and velocity is defined as the derivative of position, x, with respect to time, acceleration can be thought of as the second derivative of x with respect to t: = =.
velocity is the derivative (with respect to time) of an object's displacement (distance from the original position) acceleration is the derivative (with respect to time) of an object's velocity, that is, the second derivative (with respect to time) of an object's position. For example, if an object's position on a line is given by
Since general relativity describes four-dimensional spacetime, this represents four equations, with each one describing the second derivative of a coordinate with respect to proper time. In the case of flat space in Cartesian coordinates, we have Γ b c a = 0 {\displaystyle \Gamma _{bc}^{a}=0} , so this equation reduces to the special ...