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The instantaneous velocity equation comes from finding the limit as t approaches 0 of the average velocity. The instantaneous velocity shows the position function with respect to time. From the instantaneous velocity the instantaneous speed can be derived by getting the magnitude of the instantaneous velocity.
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.)
Instantaneous velocity of a falling object that has travelled distance on a planet with mass , with the combined radius of the planet and altitude of the falling object being , this equation is used for larger radii where is smaller than standard at the surface of Earth, but assumes a small distance of fall, so the change in is small and ...
Trajectory of a particle with initial position vector r 0 and velocity v 0, subject to constant acceleration a, all three quantities in any direction, and the position r(t) and velocity v(t) after time t. The initial position, initial velocity, and acceleration vectors need not be collinear, and the equations of motion take an almost identical ...
The instantaneous velocity of an object is the limit average velocity as the time interval approaches zero. At any particular time t , it can be calculated as the derivative of the position with respect to time: [ 2 ] v = lim Δ t → 0 Δ s Δ t = d s d t . {\displaystyle {\boldsymbol {v}}=\lim _{{\Delta t}\to 0}{\frac {\Delta {\boldsymbol {s ...
where is the instantaneous speed of the particle, the Lorentz factor, is the speed of light, and is the coordinate time. Solving for the equation of motion gives the desired formulas, which can be expressed in terms of coordinate time T {\displaystyle T} as well as proper time τ {\displaystyle \tau } .
Multiplying by the operator [S], the formula for the velocity v P takes the form: = [] + ˙ = / +, where the vector ω is the angular velocity vector obtained from the components of the matrix [Ω]; the vector / =, is the position of P relative to the origin O of the moving frame M; and = ˙, is the velocity of the origin O.
The 3-D particle tracking velocimetry (PTV) belongs to the class of whole-field velocimetry techniques used in the study of turbulent flows, allowing the determination of instantaneous velocity and vorticity distributions over two or three spatial dimensions. 3-D PTV yields a time series of instantaneous 3-component velocity vectors in the form of fluid element trajectories.