# Chapter 3 – Motion in a Straight Line

The following Topics and Sub-Topics are covered in this chapter and are available on MSVgo:

Introduction

Everything around us is in a certain kind of motion. The flow of air, the falling of leaves, the rotation of Earth, movement of vehicles, and so on – all these things represent motion. Even human beings constantly stay in motion as we stroll, run, eat, breathe, etc.

Motion is basically the change in an object’s position with respect to time. For understanding the concept of motion, you need to know about the fundamentals of displacement, velocity, and acceleration.

What is Motion in a Straight Line?

When an object moves in a straight line, it is said to be in a linear motion. This type of motion is also referred to as ‘motion in a straight line.’ In most cases, the object’s size is neglected, and it is assumed to be a point object.

#### Position, Path Length and Displacement

An object’s position is estimated using the rectangular coordinate system comprising three mutually perpendicular axes (X, Y, Z). These axes intersect at a point known as origin (O), which is the reference point. So, the position of an object is defined in terms of coordinates (x,y,z).

Linear motion involves only one-dimension, and hence, a single coordinate axis (X) is considered. When an object is in motion, its coordinates change with time. This change is measured in terms of distance, called path length. For example, if a car travelled from point O (x=0) to point A, and then from A to point B, its path length is the sum of distance OA and AB.

Since path length is a scalar quantity, it does not tell the direction of movement. Here, a vector quantity called displacement is used to reveal the direction of change in position of the object. Displacement (∆x) is calculated as the difference of initial (x1) and final positions (x2) of the object.

∆x = x2 – x1

Displacement can be positive, negative, or zero. It may or may not be equal to path length.

#### Velocity and Speed of Motion

Velocity is the measure of its change in displacement or position with the change in time. Similar to displacement, velocity is a vector quantity. The direction of an object’s velocity is the same as its direction of motion. On the other hand, speed represents the magnitude of the rate of change in distance with the change in time.

Average Velocity and Average Speed

Average velocity (vavg) refers to the ratio of displacement (∆x) and length of time interval (∆t).

vavg = ∆x/∆t

Average velocity can be negative, positive, or zero, depending upon the displacement of the object. Whereas the average speed is the ratio of total distance travelled and total time taken. The average speed is always greater than or equal to the average velocity as the total distance is always larger than or equal to the displacement. However, unlike the average velocity, the average speed does not indicate the direction of the moving object.

Instantaneous Velocity and Speed Position

As the name implies, instantaneous velocity denotes the velocity of an object at a particular instant of time. Therefore, the velocity at an instant is calculated as the limit of average velocity since the time interval is infinitesimally small. In mathematical terms, it is shown as:

v = dx/dt;

Where dx/dt is the differential quotient of displacement ‘x’ with respect to time instant ‘t’.

Further, instantaneous speed is equal to the magnitude of instantaneous velocity. Simply stated, instantaneous speed is the rate of change of distance of an object with respect to time.

#### Relative Velocity

Relative velocity is defined as the vector difference of the velocities of two objects with one of them assumed to be at rest. For instance, if two objects (P and Q) are moving along a straight line with average velocities vp and vq, respectively, the velocity of P relative to Q (vpq) is expressed as:

vpq = vp – vq

Similarly, the velocity of Q relative to P (vqp) is given by:

vqp = vq – vp

In terms of magnitude, the relative speed is the addition of average speeds of two objects, if they are moving in opposite directions. Likewise, if the two objects are travelling in the same direction, their relative speed is the difference of their average speeds.

#### Acceleration of Motion

Acceleration specifies the rate at which the velocity of an object changes with time. Being a vector quantity, the acceleration (a) is negative if the velocity decreases with time and vice-versa. It is calculated as:

a = ∆v/∆t; where ∆v is the change in velocity and ∆t is the difference in time intervals.

The concept of acceleration came into being from the study of free-falling objects. In case of a free fall, an object gains speed due to force of gravity and hence, experiences gravitational acceleration.

Motion in a Straight Line Formulas

The equations for linear motion are derived on the assumption that the object is moving under constant acceleration. These are:

• v = u + at
• s = ut + at2/2
• v2 = u2 + 2as

where ‘a’ is the constant acceleration, ‘s’ is the displacement, and ‘u’ and ‘v’ are the initial and final velocities of the object, respectively.

1. What is a straight-line motion called?

Motion along a straight line is called linear or rectilinear motion.

2. What are the types of linear motion?

There are two types of linear motion:

• Uniform motion with zero acceleration or constant velocity
• Non-uniform motion with non-zero acceleration or variable velocity

3. What is uniform motion in a straight line?

When moving along the straight line, if an object covers the equal amount of distances in the equal intervals of time, it is said to be a ‘uniform motion in a straight line’. For example, if a vehicle is travelling at a constant speed of 30 km/hr, it is covering 0.5 km per minute.

4. What is the velocity of a straight-line motion?

The velocity of a straight-line motion is called linear velocity. It refers to the rate of change of displacement with time when an object travels along a straight path (one dimension).

These basics of velocity and acceleration of motion set the foundation of many advanced principles utilized in every industry, especially in the technology, automobile, and aerospace sector.

For detailed knowledge of these concepts along with solved examples, you can check out the videos on MSVgo. This app makes learning easier via video lessons and helps in understanding concepts better.  MSVgo is an e-learning app which has been developed to embark conceptual learning in the students from grade 6-12. MSVgo has been providing the students with a core understanding of the concepts. It is a video library which is a wondrous collaboration of concepts with animations and explanatory visualisation. This app contains high-quality videos based on the curriculum, ICSE, ISC, IGCSE and IB curriculum in India. You must check out the videos on MSVgo to understand concepts in-depth on this topic.

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