We have the liberty to describe displacement vector as an independent vector ( AB) or in terms of position vectors (
Let us consider that a point object moves from point A (represented by position vector
Displacement in terms of position vectors |
---|
Thus, displacement is equal to the difference between final and initial position vectors. It is important to note that we obtain the difference between final and initial position vectors by drawing a third vector starting from the tip of the initial position vector and ending at the tip of the final position vector. This approach helps us to quickly draw the vector representing the difference of vectors and is a helpful procedural technique that can be used without any ambiguity. Equivalently, we can express displacement vector in terms of components of position vectors as :
We must emphasize here that position vectors and displacement vector are different vectors quantities. We need to investigate the relation between position vectors and displacement vector - a bit more closely. It is very important to mentally note that the difference of position vectors i.e. displacement,Δr, has different directional property to that of the position vectors themselves (
In the figure above, the position vectors
and displacement is :
This is a special case, when final position vector itself is equal to the displacement. For this reason, when motion is studied from the origin of reference or origin of reference is chosen to coincide with initial position, then displacement and final position vectors are same and denoted by the symbol, "r".
In general, however, it is the difference between final and initial position vectors, which is equal to the displacement and we refer displacement in terms of change in position vector and use the symbol Δr to represent displacement.
Magnitude of displacement is equal to the absolute value of the displacement vector. In physical sense, the magnitude of displacement is equal to the linear distance between initial and final positions along the straight line joining two positions i.e. the shortest distance between initial and final positions. This value may or may not be equal to the distance along the actual path of motion. In other words, magnitude of displacement represents the minimum value of distance between any two positions.
Example 1: Displacement
Question : Consider a person walking from point A to B to C as shown in the figure. Find distance, displacement and magnitude of displacement.
Characteristics of motion : Two dimensional
Solution : The distance covered, s, during the motion from A to C is to the sum of the lengths AB and BC.
Displacement, AC, is given as :
The displacement vector makes an angle with the x – axis given by :
and the magnitude of the displacement is :
The example above brings out nuances associated with terms used in describing motion. In particular, we see that distance and magnitude of displacement are not equal. This inequality arises due to the path of motion, which may be other than the shortest linear path between initial and final positions.
This means that distance and magnitude of displacement may not be equal. They are equal as a limiting case when particle moves in one direction without reversing direction; otherwise, distance is greater than the magnitude of displacement in most of the real time situation.
This inequality is important. It implies that displacement is not distance plus direction as may loosely be considered. As a matter of fact, displacement is shortest distance plus direction . For this reason, we need to avoid representing displacement by the symbol “s” as a vector counterpart of scalar distance, represented by “s”. In vector algebra, modulus of a vector, A, is represented by its non bold type face letter “A”. Going by this convention, if “s” and “s” represent displacement and distance respectively, then s = |s|, which is incorrect.
When a body moves in a straight line maintaining its direction (unidirectional linear motion), then magnitude of displacement, |Δr| is equal to distance, “s”. Often, this situational equality gives the impression that two quantities are always equal, which is not so. For this reason, we would be careful to write magnitude of displacement by the modulus |Δr| or in terms of displacement vector like |AB| and not by "|s|".
Example 2: Displacement
Question : Position (in meters) of a moving particle as a function of time (in second) is given by :
Find the displacement in first 2 seconds.
Characteristics of motion : Three dimensional, variable velocity, variable speed
Solution : The position vector at t = 0 and 2 seconds are calculated to identify initial and final positions. Let
When t = 0 (start of the motion)
When t = 2 s,
The displacement, Δr, is given by :
Magnitude of displacement is given by :