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Position Vector and Displacement

The position vector r of a particle P located in a plane with reference to the origin of an x-y reference frame is given by

Where x and y are components of r along x and y axes or simply they are the coordinates of the object.

Suppose a particle moves along the curve shown by the thick line and is at P at time t and P' at time t'. Then, the displacement is,

Displacement is directed from P to P'.
In component form,


The average velocity of an object is the ratio of the displacement and the corresponding time interval

Since , the direction of the average velocity is the same as that of . The velocity is given by the limiting value of the average velocity as the time interval approaches zero.

The meaning of the limiting process can be easily understood with the help of the given diagrams.

Therefore, the direction of the average velocity at any point on the path of an object is tangential to the path at that point and is in the direction of motion.

The velocity can be expressed in component form,

The magnitude of average velocity is,


The average acceleration ‘a’ of an object for a time interval Δ t moving in x-y plane is the change in velocity divided by the time interval.
a = Δ v/Δ t
a = axi+ayj

The acceleration (instantaneous acceleration) is the limiting value of the average acceleration as the time interval approaches zero.
ax = dvx/dt, ay = dvy/dt

As in the case of velocity, we can understand graphically the limiting process used in defining acceleration on a graph showing the path of the object’s motion. This is shown in the figure below. P represents the position of the object at time t and P1, P2, P3 positions after time Δ t1, Δ t2, Δ t3, respectively. The velocity vectors at points P, P1, P2, P3 are also shown in the figure. In each case of Δ t, Δ v is obtained using the triangle law of vector addition. By definition, the direction of average acceleration is the same as that of Δ v. We see that as Δ t decreases, the direction of Δ v changes and consequently, the direction of the acceleration changes.

Finally the average acceleration becomes the instantaneous acceleration and has the direction as shown. Note that in one dimension, the velocity and the acceleration of an object are always along the same straight line (either in the same direction or in the opposite direction). However, for motion in two or three dimensions, velocity and acceleration vectors may have any angle between 0o and 180o between them.

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