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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 = = =.
For example, for a changing position, its time derivative ˙ is its velocity, and its second derivative with respect to time, ¨, is its acceleration. Even higher derivatives are sometimes also used: the third derivative of position with respect to time is known as the jerk .
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.)
The second time derivative of a vector field in cylindrical coordinates is given by: ¨ = ^ (¨ ¨ ˙ ˙ ˙) + ^ (¨ + ¨ + ˙ ˙ ˙) + ^ ¨ To understand this expression, A is substituted for P , where P is the vector ( ρ , φ , z ).
As =, the first time derivatives of inside either frame, when expressed with respect to the basis of e.g. the inertial frame, coincide. Carrying out the differentiations and re-arranging some terms yields the acceleration relative to the rotating reference frame, a r {\displaystyle \mathbf {a} _{\mathrm {r} }}
An example of a generalized coordinate would be to describe the position of a pendulum using the angle of the pendulum relative to vertical, rather than by the x and y position of the pendulum. Although there may be many possible choices for generalized coordinates for a physical system, they are generally selected to simplify calculations ...
Alternatively, this same result could be obtained by computing the second time derivative of the relative position vector r B/A. [ 13 ] Assuming that the initial conditions of the position, r 0 {\displaystyle \mathbf {r} _{0}} , and velocity v 0 {\displaystyle \mathbf {v} _{0}} at time t = 0 {\displaystyle t=0} are known, the first integration ...
One may instead change to a coordinate frame fixed in the rotating body, in which the moment of inertia tensor is constant. Using a reference frame such as that at the center of mass, the frame's position drops out of the equations. In any rotating reference frame, the time derivative must be replaced so that the equation becomes