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In some applications the average velocity of an object might be needed, that is to say, the constant velocity that would provide the same resultant displacement as a variable velocity in the same time interval, v(t), over some time period Δt. Average velocity can be calculated as: [6] [7]
The linear motion can be of two types: uniform linear motion, with constant velocity (zero acceleration); and non-uniform linear motion, with variable velocity (non-zero acceleration). The motion of a particle (a point-like object) along a line can be described by its position x {\displaystyle x} , which varies with t {\displaystyle t} (time).
Every integral of motion is a constant of motion, but the converse is not true because a constant of motion may depend on time. [2] Examples of integrals of motion are the angular momentum vector, L = x × v {\displaystyle \mathbf {L} =\mathbf {x} \times \mathbf {v} } , or a Hamiltonian without time dependence, such as H ( x , v ) = 1 2 v 2 ...
Instantaneous velocity can be defined as the limit of the average velocity as the time interval shrinks to zero: = (+) (). Acceleration is to velocity as velocity is to position: it is the derivative of the velocity with respect to time.
The velocities u and u′ are the velocity of some massive object. They can also be for a third inertial frame (say F′′), in which case they must be constant. Denote either entity by X. Then X moves with velocity u relative to F, or equivalently with velocity u′ relative to F′, in turn F′ moves with velocity v relative to F.
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 ...
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These equations can be used only when acceleration is constant. If acceleration is not constant then the general calculus equations above must be used, found by integrating the definitions of position, velocity and acceleration (see above).