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Position, Path Length and Displacement

Whenever we describe an object’s motion, the first thing we talk about is the position of the object. Where is the object? It is some distance from a zero point (the point we call the origin) in a particular direction. The change in the position is what we call displacement. Let us study more about it below.

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Path Length

Distance traveled by a body is the path length. For example, if a body covers half the circumference of a circle of radius r the distance traveled is d= πr. It is a scalar quantity.

Calculating Distance in One-dimensional Motion

Total distance traveled in one-dimension can be found by adding the path lengths for all parts of motion. Note that every path length is greater than 0. Athletes race in a straight track of length 200 m and return back. The total distance traveled by each athlete is 200×2 = 400 m

Displacement Definition

Displacement of the object is equal to the length of the shortest path between the final and the initial points. Its direction is from the initial point to the final point. It is a vector quantity. For example, if a body moves along a circle of radius r and covers half the circumference, then displacement is given by s=2r.

In one-dimensional motion displacement of the object will be the shortest distance between final and initial point. For example, displacement of a particle in a circular motion would be zero when it reaches the starting point.

Displacement Formula

Displacement = final position – initial position

D =  X– Xi

D = displacement

Xf = final position

Xi = initial position

ΔX = short form for change in position

Displacement-time graph

For above graph note that displacement can be both positive and negative. Also since it is a vector, the graph is drawn for one-dimension of motion only.

Displacement-time Graph for Rest, Uniform motion and Uniform Acceleration

The graph for rest is a straight line with zero slopes. For uniform motion, the graph is a straight line with the non-zero slope. In case of a  uniform acceleration, the graph is a parabola.

Relative Displacement

It is the displacement of a point on a structure with respect to its original location or an adjacent point on the structure that has also undergone movement, can be an effective indicator of post-event structural damage.

Mathematically it is  $$\vec{r} = \vec{r}_1- \vec{r}_2$$

Distance-time graph

A distance-time graph is a graph of distance v/s time. It only lies in the first quadrant as the distance is always positive. Also, it is increasing in nature. The attached plot shows a distance-time graph.

Solved Examples For You

Q.The location of a particle has changed. What can we say about the displacement and the distance covered by the particle?

1. Neither can be zero
2. One may be zero
3. Both may be zero
4. One is positive and other is negative.

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