What is path length In Physics Class 11

The position of a body in space is denoted by its three coordinates x, y and z. To specify the position of a body, we use a reference point with respect to a set of axes.

The set of axes consists of three mutually perpendicular lines labelled as the x, y and z axes. The point of intersection of these three axes is called the origin. To specify the position of a body, its x, y and z coordinates are given with respect to this rectangular coordinate system. A stop watch is included to complete the frame of reference. When any one or more of the coordinates change with time, the body is said to be in motion with respect to the frame of reference. When the coordinates don’t change with time, the body is said to be at rest with respect to the frame of reference. Rest and motion are relative terms, and are always specified with respect to the frame of reference.

Positions to the right of O are taken as positive and to the left of O as negative. Positions above O are taken as positive and below O as negative. With time, the position of the body changes and describes a path.

The path length and displacement of a body depend on how its position changes with time. Path length is the distance traversed by a body, whereas displacement is the change in the position of the body. The path length is a scalar quantity as it has only magnitude and no direction. Since displacement has magnitude and direction, it is a vector.

The magnitude of displacement and the path length traversed by an object may or may not be the same.  There may be cases where the displacement of an object may be zero even though it has moved, but its path length will not be zero.  

The above figure gives the difference between position, path length and displacement

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

Consider the motion of a car along a straight line from the figure given below. We choose the x-axis such that it coincides with the path of the car’s motion and origin of the axis as the point from where the car started moving, i.e. the car was at x = 0 at t = 0. Let P, Q and R represent the positions of the car at different instants of time. Consider two cases of motion. In the first case, the car moves from O to P. Then the distance moved by the car is OP = +360 m. This distance is called the path length traversed by the car. In the second case, the car moves from O to P and then moves back from P to Q. During this course of motion, the path length traversed is OP + PQ = + 360 m + (+120 m) = + 480 m. Path length is a scalar quantity — a quantity that has a magnitude only and no direction (see Chapter 4).

Calculating Distance in One-dimensional Motion
Total distance travelled 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 travelled 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.
The below figure figure is an example of the following:-