3.6.1. 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 r = x ɵi + y ɵj, where x and y are components of r along x- and y- axes or simply they are the coordinates of the object.

The position vector, r, represents the location of a particle P in a two-dimensional plane. This vector is expressed in terms of its components along the x-axis (x) and y-axis (y). By using unit vectors ɵi and ɵj, which point in the positive x and y directions respectively, we can write r as r = x ɵi + y ɵj. Here, x is the horizontal coordinate, while y is the vertical coordinate, giving us a full description of the position of the particle relative to the origin (0,0) of the coordinate system.

Think about locating a point on a map: the x-coordinate could represent how far east you are from a certain reference point, while the y-coordinate might represent how far north you are. For instance, if a building is located 3 units east and 4 units north of your starting position, you could describe its location with the position vector r = 3 ɵi + 4 ɵj.

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: ∆r = r′ – r and is directed from P to P′.

Displacement is a vector quantity that describes the change in position of a particle from an initial point P at time t to a final point P′ at time t′. It is given by the equation ∆r = r′ – r, meaning that we subtract the initial position vector r from the final position vector r′. The result, ∆r, indicates not only how far the particle has moved but also in which direction it has moved from P to P′.

Imagine you walk from your home (point P) to a friend's house (point P′). The distance you walked is not the same as the displacement. Even if you walked a long path around the neighborhood, the displacement would simply be the straight-line distance from your home to your friend's house, indicating how far away you are now from your starting point, regardless of the path taken.

We can write Eq. (3.25) in a component form: ∆r = (x' - x) ɵi + (y' - y) ɵj.

In component form, the displacement vector ∆r can be expressed in terms of the changes in the x and y coordinates. Here, ∆x = x′ - x and ∆y = y′ - y represent the differences in each coordinate. Therefore, the displacement vector is written as ∆r = ∆x ɵi + ∆y ɵj, which clearly shows how much the position changes in both the x and y directions, combining them into a single displacement vector.

Returning to the earlier example of walking to a friend's house, if your home is at (1, 2) and your friend's house is at (4, 6), the change in position or displacement would be (4 - 1) in the x-direction (3 units) and (6 - 2) in the y-direction (4 units). Thus, we could express your total displacement towards your friend’s house as (3 ɵi + 4 ɵj).

The average velocity (v) of an object is the ratio of the displacement and the corresponding time interval: v = ∆r / ∆t.

Average velocity is defined as the displacement of an object divided by the time taken to achieve that displacement. The formula v = ∆r / ∆t shows how average velocity not only depends on how much distance is covered but also the time taken to cover it. A larger displacement in a shorter time results in a higher average velocity.

If you run 100 meters straight and it took you 10 seconds, your average velocity would be 10 meters per second (m/s). If you took 20 seconds to run the same distance, your average velocity would drop to 5 m/s, illustrating how both distance and time impact your average speed.

The velocity (instantaneous velocity) is given by the limiting value of the average velocity as the time interval approaches zero: v = lim (∆r/∆t) as ∆t → 0.

Instantaneous velocity represents the velocity of an object at a specific point in time. It is derived as the average velocity when the time interval approaches zero. This means looking at an object's displacement over smaller and smaller time intervals to find out exactly how fast it is moving at that very moment. The mathematical expression shows this limiting process.

If you were to watch your friend run at a track from very close, you could see how fast they are moving at any moment. Instead of measuring their speed over a longer distance, you would eventually get a very precise reading of their speed at that exact moment. This is akin to using a speedometer in a car, which shows how fast you're currently driving, rather than the average speed over the entire trip.

The average acceleration a of an object for a time interval ∆t moving in x-y plane is the change in velocity divided by the time interval: a = ∆v / ∆t.

Average acceleration quantifies how quickly the velocity of an object changes over a given time interval. The formula a = ∆v / ∆t indicates that the change in velocity (∆v) divided by the time taken (∆t) gives us the average acceleration. This can represent speeding up, slowing down, or changing direction, all of which are aspects of acceleration.

If you are driving a car and you accelerate from 0 to 60 km/h in 5 seconds, your acceleration is calculated by the change in speed (60 km/h) divided by the time taken (5 seconds), which gives you a measure of how quickly you are speeding up. If it took longer, your acceleration would be less, illustrating how time is an important factor in determining acceleration.

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