Motion in a Straight Line: Definitions, Concepts, Formulae

Motion in a Straight Line refers to the movement of objects in a straight line. Motion is an important concept in our daily lives. A car running on the road, a bird flying in the sky, or a boy walking to school, all these are examples of motion. 

A study of motion helps in the understanding of the movement of different objects. They also include the study of how fast they go and in which direction. Here we will learn about motion in a straight line, also known as rectilinear motion. We will explore fundamental concepts such as velocity, speed, acceleration, and motion under the influence of gravity.

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Motion in a Straight Line Overview

When an object moves in a straight path, it is said to be in motion in a straight line. This type of motion is also called linear or rectilinear motion. When an object travels without changing its direction, its movement is one-dimensional. Some of the examples of motion in a straight line are:

• A train is moving along a straight track.

• A car is driving straight on a highway.

• A ball rolling in a straight direction.

Position

Position refers to a point where an object is located. It is like marking the place of the object on a line. If you stand at point A and your friend at point B, both have different positions.

Distance

Distance is the total length of the path covered by an object. It shows how much ground the object has travelled, but not the direction. It is always positive.

Example: If you walk 5 meters forward and then 3 meters back, the total distance covered is 8 meters.

Displacement

Displacement is the shortest straight-line distance between the starting point and the ending point. It has both magnitude (size) and direction.
In the above example, your starting and ending positions are 2 meters apart in the forward direction. So, your displacement is 2 meters forward.

Speed and Velocity

Speed tells how fast an object moves, while velocity tells how fast and in which direction it moves.

• Speed = Distance / Time

• Velocity = Displacement / Time

Motion in a Straight Line Study Material PDF

Motion in a Straight Line is an important concept from which questions are asked in various examinations every year. The questions can be both theoretical and numerical problems. Candidates can download the Motion in a Straight Line study material from the table provided below for easy understanding of concepts and practising numerical problems related to it:

Average and Instantaneous Velocity

Velocity helps in the understanding of how quickly an object moves and in which direction it moves. It can change from one moment to another. There are two main types of velocity: average velocity and instantaneous velocity. 

• Average velocity refers to the total displacement divided by the total time taken. It shows the overall rate of motion. 

• If a bus moves 60 km north in 2 hours, its average velocity = 60 ÷ 2 = 30 km/h towards the north. Average velocity tells us the general speed of movement, but it does not describe what happens at each moment.

Instantaneous Velocity

Instantaneous velocity means the velocity of the object at a particular instant of time. It shows how fast the object is moving right now. For example, when you look at a car’s speedometer showing 50 km/h, that is its instantaneous velocity at that moment.

Difference between Average and Instantaneous Velocity:

• Average velocity is over a time period, while instantaneous velocity is at one moment.

• Average velocity gives a broad idea, while instantaneous velocity gives the exact speed at a given time.

Kinematic Equations

Kinematics is the study of motion without worrying about what causes it. In uniform acceleration (when the acceleration remains the same), three main kinematic equations are used to describe motion. These equations connect velocity (v), initial velocity (u), acceleration (a), time (t), and displacement (s).

Motion under Gravity

Gravity is the force that pulls everything towards the Earth. When we drop an object, it falls due to gravity. The motion of an object under this force is called motion under gravity. The acceleration produced by gravity is written as g.

On Earth, g = 9.8 m/s² (approximately 10 m/s² for simple calculations).

If we ignore air resistance, all objects fall at the same rate, no matter their weight. For example, if we drop a stone and a ball together, both will hit the ground at the same time.

Equations of Motion under Gravity

To derive the equations of motion under gravity, the same kinematic equations can be used by replacing a with g:

1. v = u + gt

2. s = ut + ½gt²

3. v² = u² + 2gs

When an object is thrown upwards, gravity acts in the opposite direction, so g becomes negative. When it falls downwards, g is positive.

Velocity-Time Graphs

A velocity-time graph demonstrates how the velocity of an object changes with time. It helps in understanding whether the object is speeding up, slowing down, or moving at constant speed. 

The Key parts of a Velocity-Time Graph are as follows:

• The x-axis represents time.

• The y-axis represents velocity.

• The slope (tilt) of the graph shows acceleration.

Relative Velocity in 1D

Sometimes, two objects move along the same line, and we need to find how fast one object appears to move when seen from the other. This is termed as relative velocity. In one dimension (1D), relative velocity refers to the comparison of the velocities of two moving bodies. 

If two objects A and B move along the same line, the Relative Velocity of A with respect to B (Vᴀʙ) = Vₐ - Vᵦ

Example 1:

If a bus moves at 60 km/h and a car at 80 km/h in the same direction,
Vᴀʙ = 80 – 60 = 20 km/h.
So, the car appears to move at 20 km/h faster than the bus.

Example 2:

If the same bus and car move in opposite directions,
Vᴀʙ = 80 + 60 = 140 km/h.
They appear to approach each other at 140 km/h.

Free Fall

Free fall is a special type of motion under gravity. When an object falls only due to gravity, without any other force acting on it (like air resistance), it is said to be in free fall.

• In free fall, the only acceleration acting on the object is due to gravity (g = 9.8 m/s²). The motion becomes uniform acceleration.

• When a coin is dropped from a hand, it starts from rest, and its speed increases every second as it falls. This is because gravity pulls it downward continuously.

• The equations of motion for free fall are:

1. v = u + gt

2. h = ut + ½gt²

3. v² = u² + 2gh

All objects fall at the same rate in a vacuum. The acceleration remains constant. The velocity keeps increasing till the object hits the ground.