3.1.1. Position and Displacement Vectors

To describe the position of an object moving in a plane, we need to choose a convenient point, say O as origin.

In physics, when dealing with motion in two dimensions, it's crucial to establish a reference point, called the origin. This origin serves as a starting point for measuring the position of an object. By convention, we often label this point as O. This helps in systematically defining locations in the space we are analyzing.

Think of a map of a city. The origin can be likened to a landmark like a central park or town square from which all other locations are measured. Just like you give directions from a known spot, in physics, we begin with a defined origin to describe the position of objects.

Let P and P′ be the positions of the object at time t and t′, respectively. We join O and P by a straight line. Then, OP is the position vector of the object at time t. An arrow is marked at the head of this line. It is represented by a symbol r, i.e. OP = r. Point P′ is represented by another position vector, OP′ denoted by r′.

The position of an object in a plane can be represented mathematically by vectors. When we have an object at point P at time t, we can draw a straight line from our origin O to point P; this line is known as the position vector and is denoted by r. Similarly, if the object is at a new position P′ at a later time t′, we can draw another line from O to P′, which represents the position vector r′. These vectors encapsulate both the direction and distance of the object's position from the origin.

Imagine you are in a large park. If you start at the entrance (the origin) and walk straight to the fountain (point P), you can visualize the direct path you took as a vector pointing towards the fountain. If you then walk over to a statue (point P′), you could represent your new position with another vector from the park entrance to the statue.

The length of the vector r represents the magnitude of the vector and its direction is the direction in which P lies as seen from O.

Every vector has two essential characteristics: magnitude and direction. The magnitude of a position vector refers to its length, which represents the distance between the origin O and the point P. The direction of the vector indicates where point P is located relative to the origin O. Therefore, by examining the position vector, we can understand both how far the object is from the reference point and in which direction it lies.

If we return to our park example, the distance from the entrance to the fountain gives us the magnitude, while the direction can be thought of as the compass bearing we would take to reach the fountain from the entrance. This way, we know precisely where each point is located.

If the object moves from P to P′, the vector PP′ (with tail at P and tip at P′) is called the displacement vector corresponding to motion from point P (at time t) to point P′ (at time t′).

The displacement vector describes the change in position of an object. When an object moves from point P to point P′, we can visualize this transition as a new vector, which we denote as PP′. This vector is different from the position vectors because it specifically represents how far and in what direction the object has moved from its initial position to its new position, focusing solely on the change rather than the entire path taken.

Imagine you are in a game of tag. When you start at one tree (point P) and run to another tree further away (point P′), the distance you directly cover and the straight path you take represents your displacement vector. If you took a winding path around the park, your displacement would still just account for the straight line distance between the two trees, regardless of how long your actual path was.

The displacement vector is the straight line joining the initial and final positions and does not depend on the actual path undertaken by the object between the two positions.

One of the key properties of displacement is that it is solely concerned with the initial and final positions of an object. Whether an object takes a long, winding road or a straight path to reach its destination, the displacement vector will always be drawn as a straight line connecting the starting point to the endpoint. This simplification allows us to concentrate on the overall change in position rather than the specifics of the motion itself.

Consider a road trip where you drive from one city to another. The distance you drive (the path length) might be longer if you detour around a lake or through various towns. However, your displacement merely measures the straight-line distance from the starting city to the destination city. Hence, even if your trip takes hours due to detours, your displacement remains constant based on the cities' locations.

Therefore, the magnitude of displacement is either less or equal to the path length of an object between two points. This fact was emphasised in the previous chapter also while discussing motion along a straight line.

The relationship between displacement and path length is significant in physics. The magnitude of the displacement vector will always be less than or equal to the total distance traveled (path length) by the object. This principle stems from the fact that straight-line distance (displacement) is the shortest route between two points, while the actual path length may involve various turns and deviations, lengthening the travel distance overall.

If you think about walking to a friend’s house, the straight-line distance might be only 2 kilometers. However, if you take a scenic route through the park and around other streets totaling 2.5 kilometers, your path length is longer than your displacement. Whether you prefer a straight path or a detour, both travel methods eventually connect you to your friend!

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