Vector Representations – Time and Time Intervals

Displacement and Velocity – Vector Representations

Introduction:

In this session, we will represent the position and displacement of a moving object in a graph and also learn about vector representations – time and time intervals.

Scalar and Vector Quantities

The physical quantities which have only size or magnitude and no direction are termed scalar quantities.

The physical quantities which have both size or magnitude and direction are termed vector quantities.

The examples of scalar and vector quantities are given below.

A vector is usually represented by its size along with its direction. Examples of representations of scalars and vectors are shown in the table below.

A vector quantity is represented by an arrow drawn in a frame of reference. The length of the hand represents the magnitude of the quantity and the arrow points in the direction of the quantity within that reference frame.

Position Vectors

A position vector locates an object within a frame of reference (here, a coordinate system). For example, in the picture given, the position vectors of the positions of a sprinter are shown at different instants of time when he starts running and during his run.

The coordinate system attached to the starting point allows us to designate the zero point of the variables under study and the direction in which the vectors are pointing. This establishes the origin from where the position of an object can be measured.

A position vector, ⇀d, points from the origin of a coordinate system to the location of an object at a particular instant of time.

In the adjacent picture, the position vectors of the sprinter show all positive values of their motion. Time is taken as zero when the sprinter starts running.

Non-uniform Motion

Consider the fish swimming in a water body. The swimming fishes cover unequally.

If we show the sprinter’s position before he starts racing, that would be towards the negative x-axis, as shown in the picture.

All their positions to the left, when walking towards the starting point, are considered negative. Therefore, the position and time values can be negative as well.

Displacement Vectors

The displacement of an object, ∆⇀d, is a vector that points from an initial position ⇀, d1, from which an object moves to a second position ⇀, d2, in a particular frame of reference.

The vector’s magnitude equals the straight-line distance between the two positions.

Displacement

Displacement is the vector difference between an object’s final and initial positions.

Time and Time Intervals

We usually use a clock to see the time, which is quite inconvenient in physics. In physics, the time at which an event of consideration begins is zero. Therefore, it is usually denoted as t₀ = 0 s, and other instants of time are measured with its reference and designated as t_n.

Here n represents the time at which a certain event has occurred. The time elapsed between two instants of time is called time interval, Δt.

Questions and Answers:

1. The position vector of an object is negative when ________.

Answer: The object has not started moving yet or is lying behind the starting point.

2. What is the time the moving body takes, whose data is given in the table below, to travel up to 4 m?

Answer: 51 s