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 Position Velocity Acceleration

Position

Position of a particle is its location at a given instance. In order to specifically describe the position of a particle, we need a frame of reference or a reference frame. Depending on the situation, one may use a Cartesian coordinate system or a polar coordinate system to describe the position.

Position vector is the vector drawn from the origin of the reference frame to the particle's location. In Figure 1, the position of particle P becomes

Figure 1

[1]

where i, j & k = the unit vectors, and x, y & z = the Cartesian coordinates. One may also use a polar coordinate system (Figure 2) to describe the position:

Figure 2

[2]

From [1] & [2]:

[3]

and

[4]

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Velocity

Velocity is defined as the rate of change in position or the rate of displacement. From Figure 3:

Figure 3

[5]

or

[6]

One can obtain the velocity components by taking time-derivatives of the coordinates as shown in [6]. The direction of the instantaneous velocity is along the line passing P1 and tangential to the trajectory (dotted line) in Figure 3.

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Acceleration

Acceleration is defined as the rate of change in velocity or:

[7]

From [5] and [7], it is clear that velocity is the first time-derivative of position, and acceleration is the second time-derivative of position. The first product of the motion analysis is the positions of the markers. Therefore, one can compute the velocities and accelerations of the markers through time-derivation of the position data.

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